A holomorphic functional calculus for finite families of commuting semigroups
aa r X i v : . [ m a t h . F A ] J a n A holomorphic functional calculus for finitefamilies of commuting semigroups
Jean EsterleJanuary 3, 2019
Abstract
Let A be a commutative Banach algebra such that u A 6 = { } for u ∈A \ { } which possesses dense principal ideals. The purpose of the paperis to give a general framework to define F ( − λ ∆ T , . . . , − λ k ∆ T k ) where F belongs to a natural class of holomorphic functions defined on suitableopen subsets of C k containing the "Arveson spectrum" of ( − λ ∆ T , . . . , − λ k ∆ T k ) ,where ∆ T , . . . , ∆ T k are the infinitesimal generators of commuting one-parameter semigroups of multipliers on A belonging to one of the followingclasses(1) The class of strongly continous semigroups T = ( T ( te ia ) t> suchthat ∪ t> T ( te ia ) A is dense in A , where a ∈ R . (2) The class of semigroups T = ( T ( ζ )) ζ ∈ S a,b holomorphic on an opensector S a,b such that T ( ζ ) A is dense in A for some, or equivalently for all ζ ∈ S a,b . We use the notion of quasimultiplier, introduced in 1981 by the authorat the Long Beach Conference on Banach algebras: the generators of thesemigroups under consideration will be defined as quasimultipliers on A , and for ζ in the Arveson resolvent set σ ar (∆ T ) the resolvent (∆ T − ζI ) − will be defined as a regular quasimultiplier on A , i.e. a quasimultiplier S on A such that sup n ≥ λ n k S n u k < + ∞ for some λ > and some u generating a dense ideal of A and belonging to the intersection of thedomains of S n , n ≥ . The first step consists in "normalizing" the Banach algebra A , i.e.continuously embedding A in a Banach algebra B having the same quasi-multiplier algebra as A but for which lim sup t → + k T ( te ia ) k M ( B ) < + ∞ if T belongs to the class (1), and for which lim sup ζ → ζ ∈ Sα,β k T ( ζ ) k < + ∞ for all pairs ( α, β ) such that a < α < β < b if T belongs to the class(2). Iterating this procedure this allows to consider ( λ j ∆ T j + ζI ) − as anelement of M ( B ) for ζ ∈ Res ar ( − λ j ∆ T j ) , the "Arveson resolvent set " of − λ j ∆ T j , and to use the standard integral ’resolvent formula’ even if thegiven semigroups are not bounded near the origin.A first approach to the functional calculus involves the dual G a,b ofan algebra of fast decreasing functions, described in Appendix 2. Let a = ( a , . . . , a k ) , b = ( b , . . . , b k ) , with a j ≤ b j ≤ a j + π, and de-note by M a,b the set of families ( α, β ) = ( α , β ) , . . . , ( α k , β k ) such that j = β j = a j if a j = b j and such that a j < α j ≤ β j < b j if a j < b j . Let U α,β denote the class of all functions f : ζ = ( ζ , . . . , ζ k ) → f ( ζ ) continu-ous on the product space S α,β = Π j ≤ k S α j ,β j and converging to at infin-ity such that the function σ → f ( ζ , . . . , ζ j − , η, ζ j +1 , . . . , ζ k ) is holomor-phic on S α j ,β j whenever α j < β j . Elements of the dual U ′ α,β admit a "rep-resenting measure", and we describe in appendix 1 some certainly well-known ways to implement the duality between U α,β and U ′ α,β and extendthe action of elements of U ′ α,β to vector-valued analogs spaces U α,β ( X ) and their "bounded" counterparts V α,β ( X ) via representing measures andCauchy and Fourier-Borel transforms. In appendix 2 we introduce a nat-ural algebra of fast decreasing functions, which is the intersection for ( α, β ) ∈ M a,b and z ∈ C k of all e − z U α,β , where e z ( ζ ) = e z ζ + ... + z k ζ k .The dual G a,b of this algebra is an algebra with respect to convolution,this dual space is the union for z ∈ C k and ( α, β ) ∈ M a,b of the dualspaces ( e − z U α,β ) ′ , and elements of these dual spaces act on the vector-valued spaces e − z U α,β ( X ) . This action can also be implemented via rep-resenting measures, Cauchy transforms and Fourier Borel transforms asindicated in appendix 2. If λ j S α j ,β j is contained in the domain of def-inition of T j for j ≤ k, this allows to define the action of φ ∈ G a,b on T ( λ ) = ( T ( λ . ) , . . . , T k ( λ k . ) by using the formula < T ( λ ) , φ > u = < T ( λ ζ ) . . . T k ( λ k ζ k ) u, φ ζ ,...,ζ k > ( u ∈ B ) , where B denotes a normalization of the given commutative Banachalgebra A with respect to T = ( T , . . . , T k ) , when φ ∈ ( e − z U α,β ) ′ andwhen sup ζ ∈ S α,β (cid:13)(cid:13) e zζ T ( λ ζ ) . . . T k ( λ k ζ k ) (cid:13)(cid:13) M ( B ) < + ∞ . For ( α, β ) ∈ M a,b , set S ∗ α,β = Π j ≤ k S − π − α j , π − β j . An open set U =Π æ ≤ k U j is said to be admissible with respect to ( α, β ) if for every j ≤ k the boundary ∂U j is a piecewise C -curve, if U + ǫ ⊂ U for every ǫ ∈ S ∗ α,β and if S ∗ α,β \ U is compact. Standard properties of the class H (1) ( U ) of all holomorphic functions F on U such that k F k H (1) ( U ) :=sup ǫ ∈ S ∗ α,β R ǫ + ˜ ∂U k F ( σ | dσ | < + ∞ are given in appendix 3 (when a j = b j for j ≤ k, this space is the usual Hardy space H on a product of openhalf-planes).The results of appendix 3 allow when an open set U ⊂ C k admissiblewith respect to ( α, β ) ∈ M a,b satisfies some more suitable admissibilityconditions with respect to T = ( T , . . . , T k ) and λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ to define F ( − λ ∆ T , . . . , − λ k ∆ T k ) for F ∈ H (1) ( U ) by using the formula F ( − λ ∆ T , . . . , − λ k ∆ T k )= 1(2 iπ ) k Z ǫ + ∂U F ( ζ , . . . , ζ k )( λ ∆ T + ζ I ) − . . . ( λ ∆ T k + ζ k I ) − dζ . . . dζ k , where ˜ ∂U denotes the "distinguised boundary of U and where ǫ ∈ S ∗ α,β is choosen so that ǫ + U still satisfies the required admissibility conditionswith respect to T and λ. Given T and λ , this gives a family W T,λ ofopen sets stable under finite intersection and an algebra homomorphism F → F ( − λ ∆ T , . . . , − λ k ∆ T k ) from ∪ U ∈W T,λ H (1) ( U ) into the multiplieralgebra M ( B ) ⊂ QM r ( A ) . This homomorphism extends in a natural ay to a bounded algebra homomorphism from ∪ U ∈W T,λ H ∞ ( U ) into QM r ( B ) = QM r ( A ) , and we have, if φ ∈ F α,β for some ( α, β ) ∈ M a,b , and if lim | t |→ + ∞ t ∈ Sα,β k e t z + ... + t t z k T ( t λ ) . . . T k ( t k λ k ) k = 0 for some z in thedomain of the Fourier-Borel transform of φ. FB ( φ )( − λ ∆ T , . . . , − λ k ∆ T k ) = < T ( λ ) , φ >, so that F ( − λ ∆ T , . . . , − λ k ∆ T k ) = T ( νλ j ) if F ( ζ ) = e − νζ j , when νλ j is in the domain of definition of T j . A function F ∈ H ∞ ( U ) will be said to be strongly outer if thereexists a sequence ( F n ) n ≥ of invertible elements of H ∞ ( U ) such that | F ( ζ ) | ≤ | F n ( ζ ) | and lim n → + ∞ F ( ζ ) F − n ( ζ ) = 1 for ζ ∈ U. Every boundedouter function on the open unit disc D is strongly outer, but the class ofstrongly outer functions on D k is smaller than the usual class of boundedouter functions on D k if k ≥ . We then define the Smirnov class S ( U ) tobe the class of those holomorphic functions F on U such that F G ∈ H ∞ ( U ) for some strongly outer function G ∈ H ∞ ( U ) . The boundedalgebra homomorphism F → F ( − λ ∆ T , . . . , − λ k ∆ T k ) from ∪ U ∈W T,λ into QM r ( B ) = QM r ( A ) extends to a bounded homomorphism from ∪ U ∈W T,λ S ( U ) into QM ( B ) = QM ( A ) . If F : ζ → ζ j is the j -th coordi-nate projection, then of course F ( − λ ∆ T , . . . , − λ k ∆ T k ) = λ j ∆ T j . Keywords: analytic semigroup, infinitesimal generator, resolvent, Cauchytransform, Fourier-Borel transform, Laplace transform, holomorphic functionalcalculus, Cauchy theorem, Cauchy formula
AMS classification: Primary 47D03; Secondary 46J15, 44A10
The author observed in [14] that if a Banach algebra A does not possess anynonzero idempotent then inf x ∈ A k x k≥ / k x − x k ≥ / . If x is quasinilpotent, andif k x k ≥ / , then k x k > / . Concerning (nonzero) strongly continuous semi-groups T = ( T ( t )) t> of bounded operators on a Banach space X, these elemen-tary considerations lead to the following results, obtained in 1987 by Mokhtari[24]1. If lim sup t → + k T ( t ) − T (2 t ) k < / , then the generator of the semigroupis bounded, and so lim sup k T ( t ) − T (2 t ) k = 0 .
2. If the semigroup is quasinilpotent, then k T ( t ) − T (2 t ) k > / when t issufficiently small.If the semigroup is norm continuous, and if there exists a sequence ( t n ) n ≥ of positive real numbers such that lim n → + ∞ t n = 0 and k T ( t n ) − T (2 t n ) k < / , then the closed subalgebra A T of B ( X ) generated by the semigroup possessesan exhaustive sequence of idempotents, i.e. there exists a sequence ( P n ) n ≥ ofidempotents of A T such that for every compact set K ⊂ b A T there exists n K > satisfying χ ( P n ) = 1 for χ ∈ K, n ≥ n K .3ore sophisticated arguments allowed A. Mokhtari and the author to obtainin 2002 in [18] more general results valid for every integer p ≥ . These results led the author to consider in [15] the behavior of the distance k T ( s ) − T ( t ) k for s > t near . The following results were obtained in [15]1. If there exist for some δ > two continuous functions r → t ( r ) and r → s ( r ) on [0 , δ ] , such that s (0) = 0 and such that < t ( r ) < s ( r ) and k T ( t ( r )) − T ( s ( r )) k < ( s ( r ) − t ( r )) s ( r ) s ( r ) s ( r ) − t ( r ) t ( r ) t ( r ) s ( r ) − t ( r ) for r ∈ (0 , δ ] , thenthe generator of the semigroup is bounded, and so k T ( t ) − T ( s ) k → as < t < s, s → + .
2. If the semigroup is quasinilpotent, there exists δ > such that k T ( t ) − T ( s ) k > ( s − t ) s ss − t t ts − t for < t < s ≤ δ.
3. If the semigroup is norm continuous, and if there exists two sequences ofpositive real numbers such that < t n < s n , lim n → + ∞ s n = 0 , and suchthat k T ( t n ) − T ( s n ) k < ( s n − t n ) s snsn − tnn t tnsn − tnn , then the closed subalgebra A T of B ( X ) generated by the semigroup possesses an exhaustive sequence ofidempotents.The quantities appearing in these statements are not mysterious: considerthe Hilbert space L ([0 , , and for t > define T ( t ) : L ([0 , → L ([0 , by theformula T ( t )( f )( x ) = x t f ( x ) (0 < x ≤ . Then k T ( t ) − T ( s ) k = ( s − t ) s ss − t t ts − t . This remark also shows that assertions (1) and (3) in these statements are sharp,and examples show that assertion (2) is also sharp.One can consider T ( t ) as defined by the formula R + ∞ T ( x ) dδ t ( x ) , where δ t denotes the Dirac measure at t. Heuristically, T ( t ) = e t ∆ T , where ∆ T de-notes the generator of the semigroup, and since the Laplace transform of δ t is defined by the formula L ( δ t )( z ) = R + ∞ e − zx dδ t ( x ) = e − zt , it is natural towrite L ( δ t )( − ∆ T ) = T ( t ) . More generally, if an entire function F has the form F = L ( µ ) , where µ is a measure supported by [ a, b ] , with < a < b < + ∞ , wecan set F ( − ∆ T ) = Z + ∞ T ( x ) dµ ( x ) , and consider the behavior of the semigroup near 0 in this context.I. Chalendar, J.R. Partington and the author used this point of view in [8].Denote by M c (0 , + ∞ ) the set of all measures µ supported by some interval [ a, b ] , where < a < b < + ∞ . For the sake of simplicity we restrict attention tostatements analogous to assertion 2. The following result is proved in [8]Theorem:
Let µ ∈ M c (0 , + ∞ ) be a nontrivial real measure such that R + ∞ dµ ( t ) =0 and let T = ( T ( t )) t> be a quasinilpotent semigroup of bounded operators.Thenthere exists δ > such that k F ( − s ∆ T ) k > max x ≥ | F ( x ) | for < s ≤ δ. µ = δ − δ this gives assertion 3 of Mokhtari’s result, and when µ = δ − δ p +1 this gives assertion 3 of the extension of Mokhtari’s result givenin [18] (but several variables extensions of this functional calculus would beneeded in order to obtain extensions of the results of [15]).This theory applies, for example, to quantities of the form k T ( t ) − T (2 t ) + T (3 t ) k , or Bochner integrals k R T ( tx ) dx − R T ( tx ) dx k , which are not acces-sible by the methods of [24] or [18]. Preliminary results concerning semigroupsholomorphic in a sector were obtained by I. Chalendar, J.R. Partington and theauthor in [9].More generally it would be interesting to obtain lower estimates as ( λ , . . . , λ k ) → (0 , . . . , for quantities of the form F ( − λ ∆ T , . . . , − λ k ∆ T ) when the generator ∆ T of the semigroup is unbounded, and when F is an analytic function of severalcomplex variables defined and satisfying natural growth conditions on a suitableneighbourhood of σ ar (∆ T ) , where σ ar (∆ T ) denotes the "Arveson spectrum" ofthe infinitesimal generator ∆ T of T. The purpose of the present paper is to pavethe way to such a program by defining more generally F ( − λ ∆ T , . . . , − λ k ∆ T k ) when F belongs to a suitable class of holomorphic functions on some element ofa family W T ,...,T k ,λ of open sets, and where ( T , . . . , T k ) denotes a finite familyof commuting semigroups.More precisely consider a = ( a , . . . , a k ) ∈ R k , b = ( b , . . . , b k ) ∈ R k ) satis-fying a j ≤ b j ≤ a j + π for j ≤ k, and consider a commutative Banach algebra A such that u A is dense in A for some u ∈ A and such that u A 6 = { } for u ∈ A \ { } . This allows to consider the algebra QM ( A ) of all quasimultiplierson A and the algebra QM r ( A ) of all regular quasimultipliers on A introducedby the author in [14], see section 2, and the usual algebra M ( A ) of all multipli-ers on A can be identified to the algebra of all quasimultipliers on A of domainequal to the whole of A . We will be interested here in finite families ( T , . . . , T k ) of commuting semigroups of multipliers on A satisfying the following conditions • the semigroup T j is strongly continuous on e ia j . (0 , + ∞ ) , and ∪ t> T j ( e ia j t ) A is dense in A if a j = b j , • the semigroup T j is holomorphic on the open sector S a j ,b j := { z ∈ C \{ } | a j < arg ( z ) < b j } and T j ( ζ ) A is dense in A for some (or, equivalently,for all) ζ ∈ S a j ,b j if a j < b j . The first step of the construction consists in obtaining a "normalization" A T of the Banach algebra A with respect to a strongly continuous one-parametersemigroup ( T ( t )) t> of multipliers on A . The idea behind this normalization pro-cess goes back to Feller [20], and we use for this the notion of "QM-homomorphism"between commutative Banach algebras introduced in section 2, which seemsmore appropriate than the related notion of " s -homomorphism" introduced bythe author in [14]. Set ω T = k T ( t ) k for t > . A slight improvement of aresult proved by P. Koosis and the author in section 6 of [13] shows that theweighted convolution algebra L ( R + , ω T ) possesses dense principal ideals, whichallows to construct in section 3 a commutative Banach algebra A T ⊂ QM r ( A ) A as a dense subalgebra and has dense principal ideals suchthat the injection ˜ j : QM ( A ) → QM ( A T ) associated to the norm-decreasinginclusion map j : A → A T is onto and such that ˜ j ( M ( A )) ⊂ M ( A T ) forwhich lim sup t → + k T ( t ) k M ( A T ) < + ∞ . Set φ T ( f ) = R + ∞ f ( t ) T ( t ) dt for f ∈ L ( R + , ω T ) , where the Bochner integral is computed with respect to the strongoperator topology on M ( A ) , and denote by I T the closed subalgebra of M ( A ) generated by φ T ( L ( R + , ω T )) . In section 5 we give an interpretation of the gener-ator ∆ T of the semigroup T as a quasimultiplier on I T , and we define the "Arve-son spectrum" σ ar (∆ T ) to be the set { ˜ χ (∆ T ) } χ ∈ c I T , where ˜ χ denotes the uniqueextension to QM ( I T ) of a character χ on I T , with the convention σ ar (∆ T ) = ∅ if the "Arveson ideal" I T is radical. The quasimultiplier ∆ T − λI is invertiblein QM r ( I T ) and (∆ T − λI ) − ∈ M ( A T ) ⊂ QM r ( A ) if λ ∈ C \ σ ar (∆ T ) , andwe observe in section 6 that we have, for ζ > lim sup t → + ∞ log k T ( t ) k t , (∆ T − ζI ) − = − Z + ∞ e − ζt T ( t ) dt ∈ QM ( A T ) ⊂ QM r ( A ) , which is the usual "resolvent formula" extended to strongly continuous semi-groups not necessarily bounded near the origin.In section 4 we construct a more sophisticated normalization of the Banachalgebra A with respect to a semigroup T = ( T ( ζ )) ζ ∈ S a,b which is holomorphicon an open sector S a,b , where a < b ≤ a + π. In this case the normalization A T of A with respect to the semigroup T satisfies two more conditions • T ( ζ ) u A T is dense in A T for ζ ∈ S a,b if u A is dense in A , • lim sup ζ → α ≤ arg( ζ ) ≤ β k T ( ζ ) k M ( A T ) < + ∞ for a < α < β < b. The generator of the holomorphic semigroup T is interpreted as in [7] asa quasimultiplier on the closed subalgebra of A generated by the semigroup,which is equal to the Arveson ideal I T where T denotes the restriction of T tothe half-line (0 , e i a + b . ∞ ) , and the resolvent ζ → (∆ T − ζI ) − , which is definedand holomorphic outside a closed sector of the form z + S − ie ia ,ie ib is studied insection 7.Consider again a = ( a , . . . , a k ) ∈ R k , b = ( b , . . . , b k ) ∈ R k ) satisfying a j ≤ b j ≤ a j + π for j ≤ k and a finite family T = ( T , . . . , T k ) of commuting semi-groups of multipliers on A satisfying the conditions given above. By iterating thenormalization process of A with respect to T , . . . , T k given in sections 3 and 4,we obtain a "normalization" of A with respect to the family T, see definition 8.1,which is a commutative Banach algebra B ⊂ QM r ( A ) for which the injection j : A → B is norm-decreasing, has dense range and extends to a norm-decreasinghomomorphism from M ( A ) into M ( B ) , for which the natural embedding ˜ j : QM ( A ) → QM ( B ) is onto, and for which lim sup t → + k T ( te ia j k M ( B ) < + ∞ if a j = b j , and for which lim sup ζ → αj ≤ arg ( ζ ) ≤ βj k T ( ζ ) k < + ∞ for a j < α j ≤ β j < b j if a j < b j . M a,b the set of all pairs ( α, β ) ∈ R k × R k such that α j = β j = a j if a j = b j and such that a j < α j ≤ β j < b j if a j < b j . Let W a,b be thealgebra of continuous functions f on ∪ ( α,β ) ∈ M a,b S α,β := Π j ≤ k S α j ,β j such that e z ( ζ ) f ( ζ ) → as | ζ | → in Π j ≤ k S α j ,β j for every z = ( z , . . . , z k ) ∈ C k andevery ( α, β ) ∈ M a,b , and such that the maps ζ → f ( ζ , ζ j − , ζ, ζ j +1 , . . . ζ k ) are holomorphic on S a j ,b j if a j < b j . For every element φ of the dual space G a,b = W ′ a,b there exists ( α, β ) ∈ M a,b , z ∈ C k and a measure ν of boundedvariation on S α,β := Π j ≤ k S α j ,β j such that < f, φ > = Z S α,β e zζ f ( ζ ) dν ( ζ ) ( f ∈ W a,b ) , and this formula allows to extend the action of φ to e − z V α,β ( X ) ⊃ e − z U α,β ( X ) , where X denotes a separable Banach space and where U α,β ( X ) (resp. V α,β ( X ) )denotes the algebra of continous functions f : S α,β → X which converge to 0 as ζ → ∞ (resp. bounded continuous functions f : S α,β → X ) such that the maps ζ → f ( ζ , ζ j − , ζ, ζ j +1 , . . . , ζ ) are holomorphic on S α j ,β j when α j < β j . Set U α,β := U α,β ( C ) . We describe in appendix 1 some certainly well-knownways to implement the action of U ′ α,β on V α,β ( X ) when ( α, β ) ∈ M a,b by us-ing Cauchy transforms and Fourier-Borel transforms, and these formulae areextended to the action of elements of ( e − z U α,β ) ′ to spaces e − z V α,β ( X ) in ap-pendix 2.If φ ∈ ( ∩ z ∈ C k e − z U α,β ) ′ , define the domain Dom ( FB ( φ )) of the Fourier-Borel transform FB ( φ ) of φ to be the set of all z ∈ C k such that φ ∈ ( e − z U α,β ) ′ , and set FB ( φ )( z ) = < e − z , φ > for z ∈ Dom ( FB ( φ )) . One can also definein a natural way the Fourier-Borel transform of f ∈ e − z V α,β ( X ) . Let λ ∈∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ , and set T ( λ ) ( ζ ) = T ( λ ζ , . . . , λ k ζ k ) for ζ ∈ S α,β . If lim sup | ζ |→ + ∞ ζ ∈ ˜ ∂S α,β | e − zζ |k T ( λ )( ζ ) k < + ∞ , where ˜ ∂S α,β denotes the "distinguishedboundary" of S α,β , then sup ζ ∈ S α,β (cid:13)(cid:13) e zζ T ( λ ζ ) . . . T k ( λ k ζ k ) (cid:13)(cid:13) M ( B ) < + ∞ , and one can define the action of φ on T ( λ ) by using the formula < T ( λ ) , φ > u = < T ( λ ) u, φ > = Z S α,β e zζ T ( λ ) ( ζ ) udν ( ζ ) ( u ∈ B ) , where ν is a representing measure for φe − z : f → < e − z f, φ > ( f ∈ U α,β ) . Then < T, φ > ∈ M ( B ) ⊂ QM r ( A ) . The Fourier-Borel transform of e z T ( λ ) takes values in M ( B ) and extendsanalytically to − Res ar (∆ T ( λ ) ) := Π ≤ j ≤ k ( C \ σ ar ( − λ j ∆ T j ) , which gives theformula FB ( e z T ( λ ) )( ζ ) = ( − k Π ≤ j ≤ k ( λ ∆ + ( z + ζ ) I ) − . . . ( λ k ∆ k − ( z k + ζ k I ) − . S ∗ α,β = Π j ≤ k S − π − α j , π − β j . and set W n ( ζ ) = Π ≤ j ≤ k n n + ζ j e i αj + βj ! for n ≥ , ζ = ( ζ , . . . , ζ n ) ∈ S ∗ α,β . The results of section 2 give for u ∈ B , if z ∈ Dom ( φ ) , and if lim sup | ζ |→ + ∞ ζ ∈ ˜ ∂S α,β | e − zζ |k T ( λ )( ζ ) k < + ∞ , where ˜ ∂S α,β denotesthe ”distinguished boundary" of S α,β ,< T ( λ ) , φ > u = lim ǫ → ǫ ∈ S ∗ α,β lim n → + ∞ ( − k iπ ) k Z z +˜ ∂S ∗ α,β W n ( σ − z ) FB ( φ )( σ )(( σ − ǫ ) I + λ ∆ T ) − . . . (( σ k − ǫ k ) I + λ k ∆ T k ) − udσ ! . where ˜ ∂S ∗ α,β := Π ≤ j ≤ k ∂S α j ,β j denotes the "distinguished boundary" of S ∗ α,β , and where ∂S ∗ α j ,β j is oriented from − ie iα j . ∞ to ie iβ j . ∞ . If, further, R z +˜ ∂S ∗ α,β kFB ( φ )( σ ) k| dσ | < + ∞ , then we have, for u ∈ B ,< T λ ) , φ > u = lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β < e − ǫ T ( λ ) , φ > u =lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β ( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ )(( σ − ǫ ) I + λ ∆ T ) − . . . (( σ k − ǫ k ) I + λ k A T ) − udσ. Finally, if z ∈ Dom ( φ ) , if R z +˜ ∂S ∗ α,β kFB ( φ )( σ ) k| dσ | < + ∞ , and if lim sup | ζ |→ + ∞ ζ ∈ ˜ ∂S α,β | e − zζ |k T ( λ )( ζ ) k =0 , then we have, for u ∈ B ,< T λ ) , φ > u = ( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ )( σ I + λ ∆ T ) − . . . ( σ k I + λ k A T ) − udσ. The convolution product of two elements of ( e − z U α,β ) ′ may be defined in anatural way, and if λ, φ , φ satisfies the conditions above we have < T ( λ ) , φ ∗ φ > = < T ( λ ) , φ >< T ( λ ) , φ >, but there is no direct extension of this formula to the convolution productof two arbitrary elements of G a,b , see the comments at the end of section 8.In section 9 of the paper we introduce a class U of "admissible open sets" U ,with piecewise C -boundary, of the form ( z + S ∗ α,β ) \ K , where K is bounded andwhere ( α, β ) ∈ M a,b . These open sets U have the property that U + ǫ ⊂ U for ǫ ∈ S ∗ α,β and that U + ǫ ⊂ Res ar ( − λ ∆ T ) for some ǫ ∈ S ∗ α,β . Also R ( − λ ∆ T , . ) :=( − λ ∆ T − .I ) − . . . ( − λ ∆ T k − .I ) − is bounded on the distinguished boundaryof U + ǫ for ǫ ∈ S ∗ α,β when | ǫ | is sufficiently small. Standard properties of theclass H (1) ( U ) of all holomorphic functions F on U such that k F k H (1) ( U ) :=sup ǫ ∈ S ∗ α,β R ǫ +˜ ∂U k F ( σ | dσ | < + ∞ are given in appendix 3 (when a j = b j for j ≤ k, this space is the usual Hardy space H on a product of open half-planes).8he results of appendix 3 allow when an open set U ⊂ C k admissible withrespect to ( α, β ) ∈ M a,b satisfies some more suitable admissibility conditionswith respect to T = ( T , . . . , T k ) and λ ∈ S γ,δ for some ( γ, δ ) ∈ M a − α,b − β todefine F ( − λ ∆ T , . . . , − λ k ∆ T k ) ∈ M ( B ) ⊂ QM r ( A ) for F ∈ H (1) ( U ) by usingthe formula F ( − λ ∆ T , . . . , − λ k ∆ T k )= 1(2 iπ ) k Z ǫ + ∂U F ( ζ , . . . , ζ k )( λ ∆ T + ζ I ) − . . . ( λ ∆ T k + ζ k I ) − dζ . . . dζ k , where ˜ ∂U denotes the distinguished boundary of U, where ǫ ∈ S ∗ α,β is choosenso that ǫ + U still satisfies the required admissibility conditions with respect to T and λ. Given T and λ ∈ ∪ ( α,β ) ∈ M a,b S a − α,b − β , denote by W T,λ the family of allopen sets U ⊂ C k satsisfying these admissibility conditions with respect T and λ. Then W T,λ is stable under finite intersections, ∪ U ∈W T,λ H (1) ( U ) is stableunder products and we have ( F F )( − λ ∆ T , . . . , − λ k ∆ T k ) = F ( − λ ∆ T , . . . , − λ k ∆ T k ) F ( − λ ∆ T , . . . , − λ k ∆ T k ) . This homomorphism extends in a natural way to a bounded algebra homo-morphism from ∪ U ∈W T,λ H ∞ ( U ) into QM r ( B ) = QM r ( A ) , and we have, if φ ∈ F α,β for some ( α, β ) ∈ M a,b such that λ ∈ S γ,δ for some ( γ, δ ) ∈ M a − α,b − β , and if lim | ζ |→ + ∞ ζ ∈ ˜ ∂Sα,β k e − zζ T ( λ ζ ) . . . T k ( λ k ζ k ) k = 0 for some z ∈ Dom ( FB ( φ )) , FB ( φ )( − λ ∆ T , . . . , − λ k ∆ T k ) = < T ( λ ) , φ >, so that F ( − λ ∆ T , . . . , − λ k ∆ T k ) = T ( νλ j ) if F ( ζ ) = e − νζ j , where νλ j is inthe domain of definition of T j . A function F ∈ H ∞ ( U ) will be said to be strongly outer if there exists asequence ( F n ) n ≥ of invertible elements of H ( ∞ ) ( U ) such that | F ( ζ ) | ≤ | F n ( ζ ) | and lim n → + ∞ F ( ζ ) F − n ( ζ ) = 1 for ζ ∈ U. If U is admissible with respect tosome ( α, β ) ∈ M a,b then there is a conformal map θ from D k onto U and themap F → F ◦ θ is a bijection from the set of strongly outer bounded functionson U onto the set of strongly outer bounded functions on D k . Every boundedouter function on the open unit disc D is strongly outer, but the class of stronglyouter bounded functions on D k is smaller than the usual class of bounded outerfunctions on D k if k ≥ . We then define the Smirnov class S ( U ) to be theclass of those holomorphic functions F on U such that F G ∈ H ∞ ( U ) for somestrongly outer function G ∈ H ∞ ( U ) . The bounded algebra homomorphism F → F ( − λ ∆ T , . . . , − λ k ∆ T k ) from ∪ U ∈W T,λ into QM r ( B ) = QM r ( A ) extends toa bounded homomorphism from ∪ U ∈W T,λ S ( U ) into QM ( B ) = QM ( A ) . If F : ζ → ζ j , is the j -th coordinate projection then of course F ( − λ ∆ T , . . . , − λ k ∆ T k ) = λ j ∆ T j . The author wishes to thank Isabelle Chalendar and Jonathan Partington forvaluable discussions during the preparation of this paper.9
Quasimultipliers on weakly cancellative com-mutative Banach algebras with dense principalideals
We will say that a Banach algebra A is weakly cancellative if u A 6 = { } forevery u ∈ A \ { } . In the whole paper we will consider weakly cancellative com-mutative Banach algebras with dense principal ideals, i.e. weakly cancellativecommutative Banach algebras such that the set Ω( A ) := { u ∈ A | [ u A ] − = A} is not empty.A quasimultiplier on such an algebra A 6 = { } is a closed operator S = S u/v : D S → A , where u ∈ A , v ∈ Ω( A ) , where D S := { x ∈ A | ux ∈ v A} , and where Sx is the unique y ∈ A such that vy = ux for x ∈ D S . Let QM ( A ) be the algebra of all quasimultipliers on A . A set U ⊂ QM ( A ) is said to bepseudobounded if sup S ∈ U k Su k < + ∞ for some u ∈ Ω( A ) ∩ ( ∩ S ∈ U D ( S )) , anda quasimultiplier S ∈ QM ( A ) is said to be regular if the family { λ n S n } n ≥ ispseudobounded for some λ > . The algebra of all regular quasimultipliers on A will be denoted by QM r ( A ) . A multiplier on A is a bounded linear operator S on A such that S ( uv ) = ( Su ) v for u ∈ A , v ∈ A , and the multiplier algebra M ( A ) of all multipliers on A , which is a closed subalgebra of B ( A ) , is also thealgebra of all quasimultipliers on A such that D S = A , and M ( A ) ⊂ QM r ( A ) . Also if S = S u/v ∈ QM ( A ) , w ∈ D ( S ) , R ∈ M ( A ) , then u ( Rw ) = R ( v ( Sw )) = v ( R ( Sw )) , so Rw ∈ D ( S ) , and we have R ( Sw ) = S ( Rw ) . (1)If A is unital then Ω( A ) = G ( A ) , where G ( A ) denotes the group of invertibleelements of A , and QM ( A ) = M ( A ) . The following notion if slightly more flexible than the notion of s -homomorphismintroduced by the author in [14]. Definition 2.1 : Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, and let B be a weakly cancellative Banach algebra. Ahomomorphism Φ :
A → B is said to be a QM -homomorphism if the followingconditions are satisfied(i) Φ is one-to-one, and Φ( A ) is dense in B . (ii) Φ( u ) B ⊂ Φ( A ) for some u ∈ Ω( A ) . If the conditions of definition 2.1 are satisfied, we will say that Φ is a QM -homomorphism with respect to u. Notice that Φ( u ) ∈ Ω( B ) , and so the existenceof such an homomorphism implies that B is a weakly cancellative commutativeBanach algebra with dense principal ideals. Notice also that condition (ii) showsthat B may be identified to a subalgebra of QM (Φ( A )) ≈ QM ( A ) . Proposition 2.2
Let
Φ :
A → B be a homomorphism between weakly cancella-tive commutative Banach algebras with dense principal ideals, and assume that Φ is a QM -homomorphism with respect to some u ∈ Ω( A ) . i) There exists M > such that k Φ − (Φ( u ) v ) k ≤ M k v k for v ∈ B . (ii) Φ − (Φ( u ) v ) ∈ Ω( A ) for v ∈ Ω( B ) . (iii) Set ˜Φ( S u/v ) = S Φ( u ) / Φ( v ) for S u/v ∈ QM ( A ) . Then ˜Φ : QM ( A ) →QM ( B ) is a pseudobounded isomorphism, and ˜Φ − ( S u/v ) = S Φ − (Φ( u ) u ) / Φ − (Φ( u ) v ) for S u/v ∈ QM ( B ) . Proof: (i) Set Ψ( v ) = Φ − (Φ( u ) v ) for v ∈ B . If lim n → + ∞ v n = v ∈ B , andif lim n → + ∞ ψ ( v n ) = w ∈ A , then Φ( u ) v = Φ( w ) , so that w = Ψ( v ) and (i)follows from the closed graph theorem.(ii) Let v ∈ Ω( B ) , and let ( w n ) n ≥ be a sequence of elements of A such that lim n → + ∞ v Φ( w n ) = Φ( u ) . Then lim n → + ∞ Φ − (Φ( u ) v ) w n = u ∈ Ω( A ) , andso Φ − (Φ( u ) v ) ∈ Ω( A ) . (iii) Let U ⊂ QM ( A ) be a pseudobounded set, and let w ∈ Ω( A ) ∩ ( ∩ S ∈ U D ( S )) be such that sup S ∈ U k Sw k < + ∞ . Then Φ( w ) ∈ Ω( B ) , and Φ( w ) ∈ ∩ S ∈ U D ( ˜ φ ( S )) . Since sup S ∈ U k ˜Φ( S )Φ( w ) k ≤ k Φ k sup S ∈ U k Sw k < + ∞ , this shows that ˜Φ : QM ( A ) → QM ( B ) is pseudobounded.Now set θ ( S u/v ) = S Φ − (Φ( u ) u ) / Φ − (Φ( u ) v ) = S Ψ( u ) / Ψ( v ) for S u/v ∈ QM ( B ) . It follows from (ii) that θ : QM ( B ) → QM ( A ) is well-defined. Let U ⊂ QM ( B ) be pseudobounded, and let w ∈ Ω( B ) ∩ ( ∩ S ∈ U D ( S )) be such that sup S ∈ U k Sw k < + ∞ . We have, for S = S u/v ∈ U,u ( S u/v w ) = vw, u Φ( u )( S u/v w Φ( u )) = v Φ( u ) w Φ( u ) , Φ − ( u Φ( u )) (cid:0) Φ − ( S u/v w Φ( u ) (cid:1) = Φ − ( v Φ( u ))Φ − ( w Φ( u )) . So Φ − ( wφ ( u )) ∈ D ( θ ( S )) , and θ ( S )Φ − ( w Φ( u )) = φ − (cid:0) S u/v w Φ( u ) (cid:1) . Since sup S ∈ U k ( Sw Φ( u )) k ≤ M sup S ∈ U k Sw k < + ∞ , this shows that θ : QM ( B ) →QM ( A ) is pseudobounded. We have, for S = S u/v ∈ QM ( A ) , ( θ ◦ ˜Φ)( S ) = S Φ − (Φ( u )Φ( u )) / Φ − (Φ( v )Φ( u )) = S uu /vu = S u/v = S. We also have, for S = S u/v ∈ QM ( A ) , ( ˜Φ ◦ θ )( S u/v ) = S Φ(Φ − ( u Φ( u )) / Φ(Φ − ( v Φ( u )) = S u Φ( u ) /v Φ( u ) = S u/v = S. Hence ˜Φ = QM ( A ) → QM ( B ) is bijective, and ˜Φ − = θ : QM ( B ) →QM ( A ) is pseudobounded. (cid:3) The following result is a simplified version of theorem 7.11 of [14], which willbe used in the next two sections.
Proposition 2.3
Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, and let U ⊂ QM ( A ) be a pesudobounded set stableunder products. Set L := { u ∈ ∩ S ∈ U D ( S ) | sup S ∈ U k Su k < + ∞} , and set k u k L := sup S ∈ U ∪{ I } k Su k for u ∈ L . Then ( L , k . k L ) is a Banach algebra, Ru ∈L and k Ru k L ≤ k R k M ( A ) k u k L for R ∈ M ( A ) , u ∈ L , and if we denote by B the closure of A in ( M ( L ) , k . k M ( L ) ) , then the following properties hold i) B is a weakly cancellative commutative Banach algebra, and the inclusionmap j : A → B is a QM -homomorphism with respect to w for w ∈ ( A ) ∩ L . (ii) ˜ j ( M ( A )) ⊂ M ( B ) , and k ˜ j ( R ) k M ( B ) ≤ k M ( A ) for R ∈ M ( A ) , where ˜ j : QM ( A ) → QM ( B ) is the pseudobounded isomorphism associated to j inproposition 2.2(iii).(iii) S ∈ M ( B ) , and k S k M ( B ) ≤ k S k M ( L ) ≤ for every S ∈ U. Proof: The fact that ( L , k . k L ) is a Banach space follows from a standardargument given in the proof of theorem 7.11 of [14]. Clearly, L is an ideal of A , and it follows from the definition of k . k L that k u k ≤ k u k L for u ∈ L . We have,for u ∈ L , v ∈ L , k uv k L ≤ k u k L k v k ≤ k u k L k v k L , and so ( L , k . k L ) is a Banach algebra. If R ∈ M ( A ) , u ∈ L , then it followsfrom (i) that Ru ∈ ∩ S ∈ U D ( S ) , and that we have sup S ∈ U ∪{ I } k S ( Ru ) k = sup S ∈ U ∪{ I } k R ( Su ) k ≤ k R k M ( A ) k u k L . Hence Ru ∈ L , and k Ru k L ≤ k R k M ( A ) k u k L . Now denote by B the closure of A in ( M ( L ) , k . k M ( L ) ) , and let w ∈ Ω( A ) ∩L . Since
L ⊂ B , B is weakly cancellative, and B is commutative and has denseprincipal ideals since j ( A ) is dense in B . Since w B ⊂ L ⊂ A , we see that theinclusion map j : A → B is a QM -homomorphism with respect to w, whichproves (i).Let R ∈ M ( A ) , and denote by R the restriction of R to L . Then R ∈M ( L ) , and k R k M ( L ) ≤ k R k M ( A ) . Set R u = R u for u ∈ B . Then R u ∈B , and k R u k M ( B ) ≤ k R k M ( L ) k u k M ( L ) ≤ k R k M ( A ) kk u k M ( L ) . Hence R ∈M ( B ) , and k R k M ( B ) ≤ k R k M ( A ) . Now let S ∈ U. we have, for u ∈ L , since U is stable under products, k S u k L = sup S ∈ U k S Su k ≤ sup S ∈ U k Su k = k u k L , and so S u ∈ L , and k S k M ( L ) ≤ . This implies that S ( B ) ⊂ B , so that S ∈ M ( B ) , and k S k M ( B ) ≤ k S k M ( L ) ≤ , which proves (iii). (cid:3) We have the following very easy observation.
Proposition 2.4 : Let A , A and A be weakly cancellative commutative Ba-nach algebras, and assume that Φ : A → A is a QM -homomorphism withrespect to u ∈ Ω( A ) and that Φ : A → A is a QM -homomorphism withrespect to u ∈ Ω( A ) . Then Φ ◦ Φ : A → A is a QM -homomorphism withrespect to Φ − (Φ ( u ) u ) . Proof: The homomorphism Φ ◦ Φ is one-to-one and has dense range, andit follows from proposition 2.2 (ii) that Φ − (Φ ( u ) u ) ∈ Ω( A ) . Let u ∈ A , let v ∈ A be such that Φ ( v ) = Φ ( u ) u, and let w ∈ A be such that Φ ( w ) = Φ ( u ) v. Then (Φ ◦ Φ ) (cid:0) Φ − (Φ ( u ) u ) (cid:1) u = (Φ ◦ Φ )( u )Φ ( u ) u = (Φ ( u ) v ) = (Φ ◦ Φ )( w ) ⊂ (Φ ◦ Φ )( A ) , and so Φ ◦ Φ is a QM -homomorphism with respect to Φ − (Φ ( u ) u ) . (cid:3) We will denote by b A the space of all characters on a commutative Banachalgebra A , equipped with the Gelfand topology. Recall that A is said to beradical when b A = ∅ . Definition 2.5 : Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals. For χ ∈ b A , S = S u/v ∈ QM ( A ) , set ˜ χ ( S ) = χ ( u ) χ ( v ) , and for S ∈ QM ( A ) , set σ A ( S ) := { ˜ χ ( S ) } χ ∈ b A , with the convention σ A ( S ) = ∅ if A is radical. Clearly, ˜ χ is a character on QM ( A ) for χ ∈ b A , and the map χ → ˜ χ ( S ) iscontinuous on b A for S ∈ QM ( A ) . We will use the following result in the study of the resolvent of semigroups.
Proposition 2.6
Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, and let S ∈ QM ( A ) . If λ − S has an inverse ( λ I − S ) − in QM ( A ) which belongs to A for some λ ∈ C , where I denotesthe unit element of M ( A ) , then σ A ( S ) is closed, λI − S has an inverse ( λI − S ) − in QM ( A ) which belongs to A for every λ ∈ C \ σ A ( S ) , and the A -valued map λ → ( λI − S ) − is holomorphic on C \ σ A ( S ) . Proof: If A is unital, then QM ( A ) = A , and there is nothing to prove. Soassume that A is not unital, and set ˜ A := A ⊕ C I. Then b ˜ A = { χ } ∪ { ˜ χ | ˜ A } χ ∈ b A , where χ ( a + λI ) = λ for a ∈ A , λ ∈ C . Set a = ( λ I − S ) − ∈ A . Then χ ( a ) = ˜ χ ( a ) = λ − ˜ χ ( S ) , so that ˜ χ ( S ) = λ − χ ( a ) for χ ∈ b A . Since σ A ( a ) ∪ { } = σ ˜ A is a compact subset of C , thisshows that σ A ( S ) is closed. We have, for λ ∈ C ,spec ˜ A ( I + ( λ − λ ) a ) = { } ∪ (cid:26) λ − ˜ χ ( S ) λ − ˜ χ ( S ) (cid:27) χ ∈ b A . So I + ( λ − λ ) a is invertible in ˜ A for λ ∈ C \ σ A ( S ) , and the map λ → a ( I + ( λ − λ ) a ) − ∈ A is holomorphic on C \ σ A ( S ) . We have, for λ ∈ C \ σ A ( S ) , ( λI − S ) a ( I + ( λ − λ ) a ) − = (( λ − λ ) I + ( λ − S )) a ( I + ( λ − λ ) a ) − = ( I + ( λ − λ ) a )( I + ( λ − λ ) a ) − = I. Hence λI − S has an inverse ( λI − S ) − ∈ A for λ ∈ C − σ A ( S ) , and themap λ → ( λI − S ) − = a ( I + ( λ − λ ) a ) − is holomorphic on C \ σ A ( S ) . (cid:3) Normalization of a commutative Banach alge-bra with respect to a strongly continuous semi-group of multipliers
A semigroup T = ( T ( t )) t> of multipliers on a commutative Banach algebra A is said to be strongly continuous if the map t → T ( t ) u is continuous on (0 , + ∞ ) for every u ∈ A . This implies that sup α ≤ t ≤ β k T ( t ) k < + ∞ for < α ≤ β < + ∞ , and so k T ( t ) k t has a limit ρ T as t → + ∞ , and ρ T = lim n → + ∞ k T ( n ) k n . Inthe remainder of the section T = ( T ( t )) t> will denote a strongly continuousgroup of multipliers on a weakly cancellative commutative Banach algebra A such that ∪ t> T ( t ) A is dense in A . Hence if u ∈ ∩ t> Ker ( T ( t )) , then uv = 0 for every v ∈ ∪ t> T ( t ) A , so u A = { } and u = 0 . Notice that in this situation if A has a unit element then if we set ˜ T ( t ) = T ( t ) . then ˜ T := ( ˜ T ( t )) t> is a norm-continuous semigroup of elements of A . Since ∪ t> T ( t ) A is dense in A , T ( t ) . is invertible in A for some t > , andso lim t → + k T ( t ) . − k = 0 , which implies that the generator of ˜ T is bounded.So there exist R ∈ M ( A ) ≈ A such that T ( t ) = e tR for t > if A is unital.Let ω be a positive measurable weight on (0 , + ∞ ) . Recall that if ω ( s + t ) ≤ ω ( s ) ω ( t ) for s > , t > , then the space L ω ( R + ) of all (classes of) measurablefunctions on [0 , + ∞ ) satisfying k f k ω := R + ∞ k f ( t ) | ω ( t ) dt < + ∞ , equipped withthe norm k . k ω , is a Banach algebra with respect to convolution. Set ω T ( t ) = k T ( t ) k for t > . For f ∈ L ω T ( R + ) , define Φ T ( f ) ∈ M ( A ) by the formula Φ T ( f ) u = Z + ∞ f ( t ) T ( t ) udt ( u ∈ A ) . (2)Denote by I T the closure of Φ T ( L ω T ( R + )) in M ( A ) . Let ( f n ) n ≥ be aDirac sequence, i.e. a sequence ( f n ) of nonnegative integrable functions on R + such that R + ∞ f n ( t ) dt = 1 and such that f n ( t ) = 0 a.e. on ( α n , + ∞ ) with lim n → + ∞ α n = 0 . Since the semigroup T is strongly continuous on A , a standardargument shows that lim n → + ∞ k Φ T ( f n ∗ δ t ) u − T ( t ) u k = 0 for every t > andevery u ∈ A . This shows that if v ∈ Ω( I T ) , and if w ∈ Ω( A ) , then vw ∈ Ω( A ) . The following result is then a consequence of theorem 6.8 of [13] and ofproposition 5.4 of [16].
Lemma 3.1
There exists w ∈ Ω( A ) such that lim sup t → + k T ( t ) w k < + ∞ . Proof: Let λ > log ( ρ T ) , and set ω λ ( t ) = e λt sup s ≥ t e − λs k T ( s ) k for t > . Anextension to lower semicontinuous weights of theorem 6.8 of [13] given in [16]shows that Ω( L ω λ ( R + )) = ∅ . It follows also from proposition 5.4 of [16] that k Φ T ( g ) T ( t ) k ≤ e λt k g k L ωλ for every g ∈ L ω λ ( R + ) and every t > , and that Φ T ( g ) ⊂ Ω( I T ) for every g ∈ Ω (cid:0) L ω λ ( R + ) (cid:1) . Hence if g ∈ Ω (cid:0) L ω λ ( R + ) (cid:1) and if v ∈ Ω( A ) , then Φ T ( g ) v ∈ Ω( A ) , and lim sup t → + k T ( t )Φ T ( g ) v k < + ∞ . (cid:3) The following result is a version specific to one-parameter semigroups ofproposition 2.3. 14 roposition 3.2
Let T := ( T ( t )) t> be a strongly continuous of multipliers ona weakly cancellative commutative Banach algebra A with dense principal idealssuch that ∪ t> T ( t ) A is dense in A . Let L T := { u ∈ A | lim sup t → + k T ( t ) u k < + ∞} ⊃ ∪ t> T ( t ) A , choose λ > log ( ρ T ) , set k u k λ := sup s ≥ e − λs k T ( s ) u k for u ∈ L T , with the convention T (0) = I, and set k R k λ,op := sup {k Ru k λ | u ∈L T , k u k λ ≤ } = k R k M ( L T ) for R ∈ M ( L T ) . Denote by A T the closure of A in ( M ( L T ) , k . k λ,op ) . Then ( L T , k . k λ ) is a Banach algebra, the norm topology on L T does notdepend on the choice of λ, and the following properties hold(i) The inclusion map j : A → A T is a QM -homomorphism with respectto w for every ω ∈ Ω( A ) ∩ L T , the tautological map ˜ j : S u/v → S u/v is apseudobounded isomorphism from QM ( A ) onto QM ( A T ) and if w ∈ Ω( A ) ∩L T , then ˜ j − ( S ) = S uw/vw for S = S u/v ∈ QM ( A T ) . (ii) ˜ j ( M ( A )) ⊂ M ( A T ) , and k R k M ( A T ) ≤ k R k M ( A ) for R ∈ M ( A ) . (iii) k T ( t ) k M ( A T ≤ k T ( t ) k λ,op ≤ e λt for t > , and lim sup t → + k T ( t ) u − u k λ,op = 0 for every u ∈ A T . Proof: It follows from lemma 3.1 that the family U = { e − λt T ( t ) } t> ispseudobounded for λ > log ( ρ T ) . The fact that ( L T , k . k λ ) is a Banach algebra,and assertions (i) and (ii) follow from proposition 2.2 and proposition 2.3 appliedto U , and an elementary argument given in the proof of theorem 7.1 of [16] showsthat there exists k > and K > such that k k u k λ ≤ sup ≤ t ≤ k T ( t ) u k ≤ K k u k λ for u ∈ L T , which shows that the norm topology on L T does not dependon the choice of λ. It follows also from proposition 2.3 applied to U that k T ( t ) k M ( A T ≤ k T ( t ) k λ,op ≤ e λt for t > , and lim t → + k T ( t ) u − u k λ,op = 0 for every u ∈ ∪ t> T ( t ) A . Since ∪ t> T ( t ) A T is dense in A T , a standard density argument shows that lim t → + k T ( t ) u − u k λ,op = 0 for every u ∈ A T . (cid:3) This suggests the following definition
Definition 3.3
Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, let T = (( T ( t )) t> be a strongly continuous semi-group of multipliers on A such that T ( t ) A is dense in A for t > . A normal-ization B of A with respect to T is a subalgebra B of QM ( A ) which is a Banachalgebra with respect to a norm k . k B and satisfies the following conditions(i) The inclusion map j : A → B is a QM -homomorphism, and k j ( u ) k B ≤k u k A for u ∈ A . (ii) ˜ j ( R ) ⊂ M ( B ) , and k ˜ j ( R ) k M ( B ) ≤ k R k M ( A ) for R ∈ M ( A ) , where ˜ j : QM ( A ) → QM ( B ) is the pseudobounded isomorphism associated to j intro-duced in proposition 2.2 (ii).(iii) lim sup t → + k ˜ j ( T ( t )) k M ( B ) < + ∞ . For example the algebra A T constructed in proposition 3.2 is a normalizationof the given Banach algebra A with respect to the semigroup T. Notice that if B is a normalization of A with respect to T, the same density argument as aboveshows that lim t → + k T ( t ) u − u k B = 0 for every u ∈ B . Normalization of a commutative Banach alge-bra with respect to a holomorphic semigroupof multipliers
For a < b ≤ a + π, denote by S a,b the open sector { z ∈ C \ { } | a < arg ( z ) < b } , with the convention S a,a = { re ia | r ≥ } . In this section we consider again aweakly cancellative commutative Banach algebra A with dense principal idealsand we consider a semigroup T = ( T ( ζ )) ζ ∈ S a,b of multipliers on A such that ∪ t ∈ S a,b T ( t ) A is dense in A which is holomorphic on S a,b , which implies that T ( ζ ) A is dense in A for every ζ ∈ S a,b . So T ( ζ ) u ∈ Ω( A ) for every ζ ∈ S a,b andevery u ∈ Ω( A ) . We state as a lemma the following easy observations.
Lemma 4.1
Let u ∈ M ( A ) such that lim sup ζ → ζ ∈S α,β k T ( ζ ) u k < + ∞ , where a <α ≤ β < b. (i) If λ > cos ( β − α ) (cid:16) lim r → + ∞ log ( max ( k T ( re iα ) , k T ( re iβ k ) r (cid:17) then sup ζ ∈ S α,β k e − λζe − i α + β T ( ζ ) u k < + ∞ . ( ii ) lim sup λ → + ∞ " sup ζ ∈ S α,β k e − λζe − i α + β T ( ζ ) u k = lim sup ζ → ζ ∈ Sα,β k T ( ζ ) u k . Proof: Set γ := β − α . We have, for r ≥ , r ≥ , (cid:13)(cid:13)(cid:13)(cid:13) e − λe − i α + β ( r e iα + r e iβ ) T ( r e iα + r e iβ ) u (cid:13)(cid:13)(cid:13)(cid:13) ≤ inf h e − λcos ( γ ) r k T ( r e iα ) u k e − λcos ( γ ) r k T ( r e iβ ) k , e − λcos ( γ ) r k T ( r e iα ) k e − λcos ( γ ) r k T ( r e iβ ) u k i . Set m := lim sup ζ → ζ ∈ Sα,β k T ( t ) u k , and let ǫ > . There exists δ > such that k T ( ζ ) u k ≤ m + ǫ for ≤ | ζ | ≤ δ, ζ ∈ S α,β . We obtain, considering the cases sup( r , r ) ≤ δ/ , inf( r , r ) ≤ δ/ and sup( r , r ) ≥ δ , and the case where inf( r , r ) ≥ δ , sup ζ ∈ S α,β k e − λζe i α + β T ( ζ ) u k≤ max ( m + ǫ ) sup | ζ |≤ δζ ∈ Sα,β | e − λζ | r ≥ δ/ e − λrcos ( γ ) max( k T ( re iα ) k , k T ( re iβ ) k ! , k u k sup r ≥ δ/ e − λrcos ( γ ) k T ( re iα ) k ! sup r ≥ δ/ e − λrcos ( γ ) k T ( re iβ ) k ! , and (i) and (ii) follow from this inequality. (cid:3)
16e now use a construction proposed by I. Chalendar, J.R. Partington andthe author in proposition 3.6 of [7] to associate to construct a QM - homomor-phism from A into a weakly cancellative commutative Banach algebra B suchthat sup t ∈S α,β | t |≤ k T ( t ) k < + ∞ for every α, β satisfying a < α ≤ β < b. The following result is more general than proposition 3.6 of [7].
Proposition 4.2
Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, and let T = (( T ( ζ )) ζ ∈ S a,b be a holomorphic semi-group of multipliers on A such that T ( ζ ) A is dense in A for ζ ∈ S a,b . Set α n = a + b − a n +1) , β n := b − b − a n +1) for n ≥ , and let µ = ( µ n ) n ≥ be a nonde-creasing sequence of positive real numbers satisfying the following conditions µ n > cos (cid:16) β n − α n (cid:17) lim r → + ∞ log ( max ( k T ( re iα n ) k , k ( T ( re iβ n ) k ) r ( n ≥ . (3) sup ζ ∈S αn,βn k e − µ n ζe − i a + b T ( ζ + 2 − n e i a + b ) k op,µ ≤ e − n ( µ +1) ( n ≥ . (4) where k . k op,µ is the norm on the normalization A T of A with respect to thesemigroup T := ( T ( te i ( α + β )2 )) t> associated to µ in theorem 2.2.For n ≥ set V n := { e − µ n ζe − i ( αn + βn )2 T ( ζ ) } ζ ∈S αn,βn , set W n := V . . . V n and set W = ∪ n ≥ W n , so that W is stable under products. Set L µ := { u ∈A T | sup w ∈ W k wu k µ ,op < + ∞ , and set k u k µ = sup w ∈ W ∪{ I } k wu k µ ,op for u ∈ L µ . Denote by by A µ,T the closure of A T in M ( L µ ) . Then T ( ζ ) u ∈ L µ for ζ ∈ S a,b , u ∈ A T , ( L µ , k . k µ ) is a Banach algebra, the inclusion map j : A →A µ,T is a QM homomorphism with respect to T ( ζ ) w for w ∈ Ω( A ) , and wehave the following properties(i) The tautological map ˜ j : S u/v → S u/v is a pseudobounded isomorphismfrom QM ( A ) onto QM ( A µ,T ) and if ζ ∈ S a,b then ˜ j − ( S ) = S T ( ζ ) w u/T ( ζ ) w v for S = S u/v ∈ QM ( A µ,T ) , w ∈ Ω( A ) . (ii) ˜ j ( M ( A )) ⊂ M ( A µ,T ) and k R k M ( A µ,T ) ≤ k R k M ( A ) for R ∈ M ( A ) . (iii) k T ( ζ ) k M ( A µ,T ) ≤ k T ( ζ ) k M ( L µ ) ≤ e µ n Re ( ζe − i a + b ) for ζ ∈ S α n ,β n , n ≥ , (iv) If a < α < β < b, then we have, for v ∈ A µ,T , lim sup ζ → ζ ∈ Sα,β k T ( ζ ) v − v k M ( A µ,T ) ≤ lim sup ζ → ζ ∈ Sα,β k T ( ζ ) v − v k M ( L µ ) = 0 . Proof: Since k T (2 − n ) k op,µ ≤ e − n µ , the existence of a sequence ( µ n ) n ≥ satisfying the required conditions follows from the lemma.Let t > , let n ≥ be such that − n +1 < t, and let v ∈ W. Since ∈ V n for n ≥ , we can assume that v = v . . . v n , where n ≥ n and where17 j ∈ V j for ≤ j ≤ n. Then k T (2 − j ) v j k op,µ ≤ e − j ( µ +1) for j ≥ n , and so k T (2 − n +1 − − n ) v n . . . v n k op,µ ≤ e (2 − n − − n )( µ +1) . We obtain k T (2 − n +1 ) v n . . . v n k op,µ ≤ k T (2 − n ) k op,µ k T ((2 − n +1 − − n )) v n . . . v n k op,µ ≤ e − n ( µ +1) − − n ≤ e − n ( µ +1) . Set r = t − − n n . It follows from lemma 4.1(i) that for every j ≤ n − there exists k j > such that sup v ∈ V j k T ( r ) v k op,µ ≤ k j . This gives k T ( t ) v k op,µ ≤ k T ( r ) v k op,µ . . . k T ( r ) v n − k op,µ k T (2 − n +1 ) v n . . . v n k op,µ ≤ k . . . k n − e − n ( µ +1) , and so T ( t ) A T ⊂ L µ . Now let ζ ∈ S a,b . Then ζ − te i a + b ∈ S a,b for some t > , and so T ( ζ ) A T = T ( ζ − te i ( α + β )2 ) T ( t ) A T ⊂ T ( ζ − te i ( α + β )2 ) L µ ⊂ L µ . Since W is stable under products, the fact that L µ is a Banach algebra followsfrom proposition 2.3, which also implies (ii) and (iii). Let ζ ∈ S a,b . Since (iv)holds for u ∈ T ( ζ ) A µ,T , and since T ( ζ ) A µ,T ) is dense in ( A µ,T , k . k L µ ) (iv)follows from (iii) by a standard density argument.Let ζ ∈ S a,b , and let w ∈ Ω( A ) ⊂ Ω( A T ) . Since T ( ζ/ w ∈ Ω( A ) , andsince lim sup t → + k T ( t ) T ( ζ ) w k < + ∞ , it follows from proposition 2.3 thatthe inclusion map j : A → A T is a QM -homomorphism with respect to T ( ζ/ w. Since T ( ζ/ w ⊂ M µ ∩ Ω( A T ) , it follows also from proposition2.3 that the inclusion map j : A T → A µ,T is a QM -homomorphism withrespect to T ( ζ/ w, and it follows then from proposition 2.4 that the inclu-sion map j = j ◦ j : A → A µ,T is a QM homomorphism with respectto T ( ζ/ wT ( ζ/ w = T ( ζ ) w . It follows then from proposition 2.2 that thetautological map ˜ j : S u/v → S u/v is a pseudobounded homomorphism from QM ( A ) onto QM ( A µ,T ) and that ˜ j − ( S u/v ) = S T ( ζ ) uw /T ( ζ ) vw for S = S u/v ∈QM ( A µ,T ) . Since T ( ζ ) A µ,T ⊂ T ( ζ/
2) [ T ( ζ/ A µ,T ] ⊂ T ( ζ/ A T ⊂ A , andsince T ( ζ ) v A contains T ( ζ ) vw A which is dense in A for v ∈ Ω( A µ,T ) , we have T ( ζ ) v ∈ Ω( A ) for ζ ∈ S a,b , v ∈ Ω( A µ,T ) , and so ˜ j − ( S ) = S T ( ζ ) uw /T ( ζ ) vw for S = S u/v ∈ QM ( A µ,T ) . (cid:3) We will use the following notion.
Definition 4.3
Let A be a weakly commutative Banach algebra with dense prin-cipal ideals, and let T := ( T ( ζ )) ζ ∈ S a,b be an analytic semigroup of multipliers on A such that T ( ζ ) A is dense in A for ζ ∈ S a,b . A normalization of the algebra A with respect to the semigroup T is a subalgebra B of QM ( A ) which is a Banachalgebra with respect to a norm k . k B and satisfies the following conditions(i) There exists u ∈ Ω( A ) such that the inclusion map j : A → B is a QM -homomorphism with respect to T ( ζ ) u for every ζ ∈ S a,b , and k j ( u ) k B ≤ k u k A for u ∈ A . ii) ˜ j ( R ) ⊂ M ( B ) , and k ˜ j ( R ) k M ( B ) ≤ k R k M ( A ) for R ∈ M ( A ) , where ˜ j : QM ( A ) → QM ( B ) is the pseudobounded isomorphism associated to j intro-duced in proposition 2.2 (ii).(iii) lim sup t → ζ ∈ Sα,β k ˜ j ( T ( t )) k M ( B ) < + ∞ for a < α < β < b. If B is a normalization of A with respect to the holomorphic semigroup T = ( T ( ζ )) ζ ∈ S a,b , a standard density argument shows that if a < α < β < b then lim sup ζ → ζinSα,β k T ( ζ ) u − u k B = 0 for every u ∈ B Notice that the algebra A ,T = A µ , T and its norm topology associated tothe norm k . k µ ,op discussed above do not depend on the choice of µ . This isno longer the case for the Banach algebra A µ,T and its norm topology, whichmay depend on the choice of the sequence µ. In order to get a more intrin-sic renormalization one could consider the Fréchet algebra L := ∩ n ≥ { u ∈A T | sup ζ ∈ Sαn,βn k u k µ ,op ≤ k T ( ζ ) u k µ ,op < + ∞} , then consider the closed subalgebra U of L generated by the semigroup and introduce an intrinsic normalization of A T to be the closure of U in M ( U ) with respect to the Mackey-convergence asso-ciated to a suitable notion of boundedness on subsets of M ( U ) , but it seemsmore convenient to adopt the point of view used in proposition 4.2. In this section we consider again a weakly cancellative commutative Banachalgebra A with dense principal ideals and a strongly continuous semigroup T =( T ( t )) t> of multipliers on A such that ∪ t> T ( t ) A is dense in A . We set again ω T ( t ) = k T ( t ) k for t > . Denote by M ω T the space of all measures ν on (0 , + ∞ ) such that R + ∞ ω T ( t ) d | ν | ( t ) < + ∞ , and for ν ∈ M ω T define a Φ T : M ω T →M ( A ) by the formula Φ T ( ν ) u := Z + ∞ T ( t ) udν ( t ) ( u ∈ A ) . (2 ′ ) The Bochner integral is well-defined since the semigroup is strongly contin-uous, M ω T is a Banach algebra with respect to convolution of measures on thehalf-line, and we will identify again the space L ω T of (classes of) measurablefunctions on [0 , + ∞ ) satisfying R + ∞ | f ( t ) |k T ( t ) k dt < + ∞ to the ideal of all ν ∈ M ω T which are absolutely continuous with respect to Lebesgue measure.Denote by B T the closure of φ T ( M ω ) in M ( A ) , and denote again by I T theclosure of Φ T ( L ω T ) in M ( A ) , so that the "Arveson ideal" I T is a closed idealof B T . The idea of considering the generator of a semigroup as a quasimultiplier onsome suitable Banach algebra goes back to [21] and [22] for groups of bounded19perators and, more generally, for groups of regular quasimultipliers. An ob-vious such interpretation was given by I. Chalendar, J. R. Partington and theauthor in [7] for analytic semigroups, and the author interpreted in section 8 of[16] the generator of a semigroup of bounded operators which is weakly continu-ous in the sense of Arveson [2] as a quasimultiplier on the corresponding Arvesonideal I T . Since in the present context Φ T (cid:0) Ω( L ω T ) (cid:1) ⊂ Ω( I T ) and uv ∈ Ω( A ) for u ∈ Ω( I T ) , v ∈ Ω( A ) , the map j T : S u/v → S uw/vw is a pseudobounded homo-morphism from QM ( I T ) into QM ( A ) for every w ∈ Ω( A ) , and the definitionof j T does not depend on the choice of w. The generator ∆ T, I T of T considered as a strongly continuous semigroup ofmultipliers on I T has been defined in [16], def. 8.1 by the formula ∆ T, I T = S − Φ T ( f ′ ) / Φ T ( f ) , (5)where f ∈ C ([0 , + ∞ )) ∩ Ω (cid:0) L ω T (cid:1) satisfies f = 0 , f ′ ∈ L ω T , and an easy verification given in [16] shows that this definition does notdepend on the choice of f . This suggests the following definition
Definition 5.1 : The infinitesimal generator ∆ T, A of T is the quasimultiplieron A defined by the formula ∆ T, A = j T (∆ T, I T ) = S − Φ T ( f ′ ) u / Φ T ( f ) u , where f ∈ C ([0 , + ∞ )) ∩ Ω (cid:0) L ω T (cid:1) satisfies f = 0 , f ′ ∈ L ω T , and where u ∈ Ω( A ) . Assume that f and u also satisfy the conditions of the definition, and set f = f ∗ f , u = u u Since Ω (cid:0) L ω T (cid:1) is stable under convolution, f ∈ Ω (cid:0) L ω T (cid:1) , and f ′ = f ′ ∗ f = f ∗ f ′ ∈ L ω T is continuous. Also f (0) = 0 , and we have Φ T ( f ′ ) u Φ T ( f ) u = Φ T ( f ′ )Φ T ( f ) u Φ T ( f ) u = Φ T ( f ′ ) u Φ T ( f ) u and similarly Φ T ( f ′ ) u Φ T ( f ) u = Φ T ( f ′ ) u Φ T ( f ) u , which shows thatthe definition of ∆ T , A does not depend on the choice of f and u . Proposition 5.2
Let T and T be two semigroups satisfying the conditionsof definition 5.1with respect to A . If T ( t ) T ( t ) = T ( t ) T ( t ) for t > , then ∆ T T , A = ∆ T , A + ∆ T , A . Proof: Let f and f be two functions on [0 , + ∞ ) satisfying the conditionsof definition 5.1 with respect to T and T . Since ω T T ( t ) ≤ ω T ( t ) ω T ( t ) for t > , f f satisfies the conditions of definition 5.1 with respect to T T , and itfollows from Leibnitz rule and Fubini’s theorem that we have Φ T T ( f f ) = Φ T ( f )Φ T ( f ) , Φ T T (( f f ) ′ ) = Φ T ( f ′ )Φ T ( f )+Φ T ( f )Φ T ( f ′ ) , and the results follows. (cid:3)
20e now give a link between the quasimultiplier approach and the classicalapproach based on the study of T ( t ) u − ut as t → + . A proof of the followingfolklore result is given for example in [16], lemma 8.4.
Lemma 5.3
Let ω be a lower semicontinuous submultiplicative weight on (0 , + ∞ ) , and let f ∈ C ([0 , + ∞ )) ∩ L ω . If f (0) = 0 , and if f ′ ∈ L ω , then the Bochnerintegral R + ∞ t ( f ′ ∗ δ s ) ds is well-defined in L ω for t ≥ , and we have f ∗ δ t − f = − Z t ( f ′ ∗ δ s ) ds, and lim t → + (cid:13)(cid:13)(cid:13)(cid:13) f ∗ δ t − ft + f ′ (cid:13)(cid:13)(cid:13)(cid:13) L ω = 0 . It follows from the lemma that we have if f ∈ C ([0 , + ∞ )) ∩ L ω T , and if f ′ ∈ L ω T , T ( t )Φ T ( f ) − Φ T ( f ) = − Z t T ( s )Φ T ( f ′ ) ds ( t ≥ . (6) Proposition 5.4 (i) Let u ∈ A . If lim t → + k T ( t ) u − ut − v k = 0 for some v ∈ A , then u ∈ D ∆ T, A , and ∆ T, A u = v. (ii) Conversely if lim sup t → + k T ( t ) k < + ∞ , then lim sup t → + k T ( t ) u − ut − ∆ T, A u k = 0 for every u ∈ D ∆ T, A . Proof: (i) If u ∈ A , and if lim t → + k T ( t ) u − ut − v k = 0 for some v ∈ U T , let f ∈ C ([0 , + ∞ )) ∩ Ω (cid:0) L ω T (cid:1) satisfiying f = 0 , f ′ ∈ L ω T , and let u ∈ Ω( A ) . Itfollows from the lemma that we have, with respect to the norm topology on A , − Φ T ( f ′ ) u u = (cid:20) lim t → + T ( t )Φ T ( f ) − Φ T ( f ) t (cid:21) u u = Φ T ( f ) u (cid:20) lim t → + T ( t ) u − ut (cid:21) = Φ T ( f ) u v, and so u ∈ D ∆ T, A , and ∆ T, A u = v. (ii) Conversely assume that lim sup t → + k T ( t ) k < + ∞ , let u ∈ D ∆ T, A , let f ∈ C ([0 , + ∞ )) ∩ Ω (cid:0) L ω T (cid:1) such that f ′ ∈ L ω T , and let u ∈ Ω( A ) . Let v =∆ T , I T, B u. It follows from (6) that we have, for t ≥ , Φ T ( f ) u Z t T ( s ) vds = Z t T ( s )Φ T ( f ) u vds = − Z t T ( s ) φ T ( f ′ ) u uds = − (cid:20)Z t T ( s )Φ T ( f ′ ) ds (cid:21) u u = [ T ( t )Φ T ( f ) u − Φ T ( f ) u ] u = Φ T ( f ) u ( T ( t ) u − u ) . Since Φ T ( f ) u ∈ Ω( A ) , this shows that T ( t ) u − u = R t T ( s ) vds, and so lim t → + (cid:13)(cid:13)(cid:13) T ( t ) u − ut − v (cid:13)(cid:13)(cid:13) = 0 . (cid:3) We now consider a normalization A with respect to T , see definition 3.3.21 roposition 5.5 Let B be a normalization of A with respect to T. Set v λ ( t ) = te − λt for λ ∈ R , t ≥ , and let λ > log ( ρ T ) . (i) If u ∈ Ω( B ) , then ∆ T, B = − S R + ∞ v ′ λ ( t ) T ( t ) udt/ R + ∞ v λ ( t ) T ( t ) udt . (ii) Let ˜ j : QM ( A ) → QM ( B ) be the pseudobounded isomorphism givenin proposition 3.2 (i). Then ˜ j − (∆ T, B ) = ∆ T, A . So if u ∈ Ω( A ) , and if lim sup t → + k T ( t ) u k < + ∞ , then ∆ T, A = − S R + ∞ v ′ λ ( t ) T ( t ) udt/ R + ∞ v λ ( t ) T ( t ) udt . Proof: (i) Set e λ ( t ) = e λt for λ ∈ R , t ≥ . If λ > µ > log ( ρ T ) , then v λ ∈ L e µ ⊂ L ω T , where ˜ ω T ( t ) = k T ( t ) k M ( A T ) , and L e µ is dense in L ω T sinceit contains the characteristic function of [ α, β ] for < α < β < + ∞ . It followsfor example from Nyman’s theorem [25] about closed ideals of L ( R + ) that v λ e µ ∈ Ω( L ( R + )) and so v λ ∈ Ω( L e µ ) ⊂ Ω( L ω T ) . So v λ and u satisfy theconditions of definition 5.1 with respect to T and B , and (i) holds.(ii) The map ˜ j is the tautological map S u/v → S u/v , where u ∈ A ⊂ B and v ∈ Ω( A ) ⊂ Ω( B ) . Now let f ∈ L ω T ∩ C ([0 , + ∞ )) satisfying definition5.1 with respect to T and A and let u ∈ Ω( A ) . Since Ω( L ω T ) ⊂ Ω( L ω T ) , andsince Ω( A ) ⊂ Ω( B ) , it follows from definition 5.1 that ˜ j (∆ T, A ) = ∆ T, B , and so ˜ j − (∆ T, B ) = ∆ T, A . Let u ∈ Ω( A ) ⊂ Ω( B ) , and assume that lim sup t → + k T ( t ) u k < + ∞ . Let w ∈ Ω( A ) be such that w B ⊂ A . Since v λ ∈ Ω( L ω λ ) , we see as in the proofof lemma 3.1 that R + ∞ v λ ( t ) T ( t ) udt ∈ Ω( B ) , and it follows from proposition2.2 that w R + ∞ v λ ( t ) T ( t ) udt ∈ Ω( A ) . Using the characterization of ˜ j − given inproposition 2.2, we obtain ∆ T, B = − S R + ∞ v ′ λ ( t ) T ( t ) udt/ R + ∞ v λ ( t ) T ( t ) udt , and ∆ T, A = ˜ j − (∆ T, B ) = − S w R + ∞ v ′ λ ( t ) T ( t ) udt/w R + ∞ v λ ( t ) T ( t ) udt = − S R + ∞ v ′ λ ( t ) T ( t ) udt/ R + ∞ v λ ( t ) T ( t ) udt . (cid:3) We will denote by c I T the space of characters of I T , equipped with the usualGelfand topology. Notice that if χ ∈ c I T then there exists a unique character ˜ χ on QM ( I T ) such that ˜ χ | I T = χ, which is defined by the formula ˜ χ ( S u/v ) = χ ( u ) χ ( v ) for u ∈ I T , v ∈ Ω( I T ) . Definition 5.6
Assume that I T is not radical, and let S ∈ QM ( I T ) . TheArveson spectrum σ ar ( S ) is defined by the formula σ ar ( S ) = { λ = ˜ χ ( S ) : χ ∈ c I T } . ν is a measure on [0 , + ∞ ) , the Laplace tranform of ν is defined by theusual formula L ( ν )( z ) = R + ∞ e − zt dν ( t ) when R + ∞ e − Re ( z ) t d | ν | ( t ) < + ∞ . We have the following easy observation.
Proposition 5.7
Let ν ∈ M ω T . Then we have, for χ ∈ c I T , ˜ χ (cid:18)Z + ∞ T ( t ) dν ( t ) (cid:19) = L ( ν )( − ˜ χ (∆ T, I T )) . (7) Similarly we have, for ν ∈ M ˜ ω T , χ ∈ c I T , ˜ χ (Φ T ( ν )) = ˜ χ (cid:18)Z + ∞ T ( t ) dν ( t ) (cid:19) = L ( ν )( − ˜ χ (∆ T, I T )) = L ( ν )( − ˜ χ (∆ T , I T, B )) . (8) In particular ˜ χ ( T ( t )) = e ˜ χ (∆ T, I T ) t for t > . Proof: If χ ∈ c I T , then ˜ χ | A T is a character on A T , the map t → ˜ χ ( T ( t )) iscontinuous on (0 , + ∞ ) and so there exists λ ∈ C such that ˜ χ ( T ( t )) = e − λt for t > , and | e − λt | ≤ k T ( t ) k , which shows that Re ( λ ) ≥ − log ( ρ T ) . Let u ∈ Ω( I T ) , and let ν ∈ M ω T . We have χ ( u ) ˜ χ (cid:18)Z + ∞ T ( t ) dν ( t ) (cid:19) = χ (cid:18) u Z + ∞ T ( t ) dν ( t ) (cid:19) = χ (cid:18)Z + ∞ T ( t ) udν ( t ) (cid:19) = Z + ∞ χ ( T ( t ) u ) dν ( t ) = χ ( u ) Z + ∞ e − λt dν ( t ) = χ ( u ) L ( ν )( λ ) , and so ˜ χ (Φ T ( ν )) = L ( ν )( λ ) . Let f ∈ C ((0 , + ∞ )) ∩ Ω( I T ) such that f (0) = 0 . We have λ L ( f )( λ ) = L ( f ′ )( λ ) = χ (Φ T ( f ′ )) = − ˜ χ (∆ T, I T Φ T ( f ))= − ˜ χ (∆ T, I T ) χ ( φ T ( f )) = − ˜ χ (∆ T, I T ) L ( f )( λ ) , and so λ = − ˜ χ (∆ T, I T ) , which proves (7), and formula (8) follows from asimilar argument. In particular χ ( T ( t )) = L (∆ t )( − ˜ χ (∆ T , I T )) = e ˜ χ (∆ T, I T ) t for t > . (cid:3) The following consequence of proposition 5.7 pertains to folklore.
Corollary 5.8
Assume that I T is not radical. Then the map χ → ˜ χ (∆ T, I T ) is ahomeomorphism from c I T onto σ ar (∆ T, I T ) , and the set Λ t := { λ ∈ σ ar (∆ T, I T ) | Re ( λ ) ≤ t } is compact for every t ∈ R , so that σ ar (∆ T, I T ) is closed. Proof: Let f ∈ C ((0 , + ∞ )) ∩ Ω( I T ) such that f (0) = 0 . We have χ (Φ T ( f )) =0 and ˜ χ (∆ T, I T ) = − χ (Φ T ( f ′ )) χ (Φ T ( f )) for χ ∈ c I T , and so the map χ → ˜ χ (∆ T ) is con-tinuous with respect to the Gelfand topology on c I T . f ∈ L ω T . It follows from proposition 5.6 that we have, for χ ∈ c I T , χ (Φ T ( f )) = L ( f )( − ˜ χ (∆ T, I T )) . Since the set { u = Φ T ( f ) : f ∈ L ω T } is dense in I T , this shows that the map χ → ˜ χ (∆ T, I T ) is one-to-one on c I T , and that the inverse map σ ar (∆ T, I T ) → c I T is continuous with respect to the Gelfand topology.Now let t ∈ R , and set U t := { χ ∈ c I T : Re ( χ (∆ T , I T )) ≤ t } . Then | ˜ χ ( T (1)) | ≥ e − t for χ ∈ U t , and so does not belong to the closure of U t with respect to theweak ∗ topology on the unit ball of the dual of I T . Since c I T ∪ { } is compactwith respect to this topology, U t is a compact subset of c I T , and so the set Λ t iscompact, which implies that σ ar (∆ T, I T ) = ∪ n ≥ Λ n is closed. (cid:3) We now wish to discuss the resolvent of the generator of a strongly continuoussemigroup T = ( T ( t )) t> of multipliers on A , where A is a weakly cancellativecommutative Banach algebra with dense principal ideals, and where ∪ t> T ( t ) A is dense in A . From now on we will write ∆ T = ∆ T, A and we will denote by D ∆ T , A the domain of ∆ T considered as a quasimultiplier on A . The Arvesonideal I T is as above the closed subalgebra of M ( A ) generated by Φ T ( L ω T ) . The Arveson resolvent set is defined by the formula
Res ar (∆ T, I T ) = C \ σ ar (∆ T, I T ) , with the convention σ ar (∆ T, I T ) = ∅ if I T is radical. The usual"resolvent formula," interpreted in terms of quasimultipliers, shows that λI − ∆ T, I T ∈ QM ( I T ) is invertible in QM ( I T ) and that its inverse ( λI − ∆ T, I T ) − belongs to the Banach algebra I T, B ⊂ QM r ( I T ) obtained by applying theorem2.2 to I T with respect to the semigroup T, and that we have, for Re ( λ ) >log ( ρ T ) , ( λI − ∆ T, I T ) − = Z + ∞ e − λs T ( s ) ds ∈ I T, B , where the Bochner integral is computed with respect to the strong oper-ator topology on M ( I T, B ) . Also the I T, B -valued map λ → ( λI − ∆ T, I T ) − is holomorphic on Res ar ( T, I T ) . The details of the adaptation to the contextof quasimultipliers of this classical part of semigroup theory are given in [16],proposition 10.2.We now give a slightly more general version of this result, which applies inparticular to the case where B is the normalization A T of A with respect to thesemigroup T introduced in proposition 2.2.In the following we will identify the algebras QM ( A ) and QM ( B ) using theisomorphism ˜ j intoduced in proposition 2.2 (iii) if B is a normalization of A with respect to T. We set Φ T, B ( ν ) u = R + ∞ T ( t ) udν ( t ) for u ∈ B , ν ∈ M ω T, B , where ω T, B ( t ) = k T ( t ) k M ( B ) for t > , and we denote by I T, B the closure of Φ T, B ( L ω T, B ) in M ( B ) . roposition 6.1 Let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, let T = (( T ( t )) t> be a strongly continuous semi-group of multipliers on A such that T ( t ) A is dense in A for t > , and let B be a normalization of A with respect to T. Set
Res ar (∆ T ) = Res ar (∆ T, I T ) = C \ σ ar (∆ T, I T ) . The quasimultiplier λI − ∆ T ∈ Q M ( A ) admits an inverse ( λI − ∆ T ) − ∈I T, B ⊂ M ( B ) ⊂ Q M r ( A ) for λ ∈ Res ar (∆ T ) , and the map λ → ( λI − ∆ T ) − isan holomorphic map from Res ar (∆ T ) into I T, B . Moreover we have, for Re ( λ ) >log ( ρ T ) , ( λI − ∆ T ) − = Z + ∞ e − λs T ( s ) ds ∈ I T, B , where the Bochner integral is computed with respect to the strong operatortopology on M ( B ) , and k ( λI − ∆ T ) − k M ( B ) ≤ R + ∞ e − Re ( λ ) t k T ( t ) k M ( B ) dt. Proof: We could deduce this version of the resolvent formula from proposi-tion 10.2 of [16], but we give a proof for the sake of completeness. Set again e λ ( t ) = e λt for t ≥ , λ ∈ C . Assume that Re ( λ ) > log ( ρ T ) ≥ lim t → + ∞ log k T ( t ) k M ( B ) t , let v ∈ I T, B , and set a = Φ T, B ( e − λ ) . We have av = Z + ∞ e − λs T ( s ) vds, T ( t ) av − av = Z + ∞ e − λs T ( s + t ) vds − Z + ∞ e − λs T ( s ) vds = e λt Z + ∞ t e − λs T ( s ) vds − Z + ∞ e − λs T ( s ) vds = ( e λt − av − e λt Z t e − λs T ( s ) vds. Since lim t → + k T ( t ) v − v k I T, B = 0 , we obtain lim t → + (cid:13)(cid:13)(cid:13)(cid:13) T ( t ) av − avt − λav + v (cid:13)(cid:13)(cid:13)(cid:13) I T, B = 0 , and so av ∈ D ∆ T , I T, B , and ∆ T, I T, B ( av ) = λav − v. This shows that a I T, B ⊂D ∆ T, I T, B , and that ( λI − ∆ T, I T, B ) av = v for every v ∈ I T, B . We have λI − ∆ T, I T, B = S u/v , where u ∈ I T, B , v ∈ Ω( I T, B ) , and we see that ua = v. Hence u ∈ Ω( I T, B ) , λI − ∆ T, I T, B is invertible in QM ( I T, B ) , and ( λI − ∆ T, B ) − = a =Φ T, B ( e − λ ) = R + ∞ e − λt T ( t ) dt ∈ I T, B , where the Bochner integral is computedwith respect to the strong operator topology on M ( I T, B ) . Let χ ∈ d I T, B . Then χ ◦ ˜ j ∈ c I T , and so σ ar (∆ T, I T, B ) ⊂ σ ar (∆ T, I T ) . It followsthen from proposition 2.6 that λI − ∆ T, B has an inverse ( λI − ∆ T, B ) − ∈ I T, B in QM r ( I T, B ) for λ ∈ Res ar (∆ T, I T ) and that the I T, B -valued map λ → ( λI − ∆ T, B ) − is holomorphic on Res ar (∆ T, B ) . Fix u ∈ Ω( A ) ⊂ Ω( B ) , and set j T ( S ) = S uu /vu for S = S u/v ∈ QM ( I T, B ) . Then j T : QM ( I T, B ) → QM ( B ) is a pseudobounded homomorphism, and j T (∆ T, I T, B ) = ∆ T, B . Identifying I T, B to a subset of QM ( I T, B ) as above inthe obvious way, we see that the restriction of j T to I T, B is the identity map,and so λI − ∆ T is invertible in QM ( B ) for λ ∈ Res ar (∆ T ) , ( λI − ∆ T ) − = λI − ∆ T, I T, B ) − and the I T, B -valued map λ → ( λI − ∆ T, B ) − is holomorphicon Res ar (∆ T ) . If Re ( λ ) > log ( ρ T ) ≥ lim t → + ∞ log k T ( t ) k M ( B ) t , then if u ∈ I T, B , v ∈ B , wehave ( λI − ∆ T ) − uv = (( λI − ∆ T, B ) − u ) v = Z + ∞ e − λt T ( t ) uvdt. Since uv ∈ Ω( B ) for u ∈ Ω( I T, B ) , v ∈ Ω( B ) , u B is dense in B for u ∈ Ω( I T, B ) , and we obtain ( λI − ∆ T ) − = R + ∞ e − λt T ( t ) dt ∈ I T, B , where the Bochnerintegral is computed with respect to the strong operator topology on M ( B ) , sothat k ( λI − ∆ T ) − k M ( B ) ≤ R + ∞ e − Re ( λ ) t k T ( t ) k M ( B ) dt. (cid:3) If we consider ∆ T as a quasimultiplier on B , the fact that ( λI − ∆ T ) − ∈M ( B ) is the inverse of λI − ∆ T for λ ∈ Res (∆ T ) means that ( λI − ∆ T ) − v ∈D ∆ T, B and that ( λI − ∆ T ) (cid:0) ( λI − ∆ T ) − v (cid:1) = v for every v ∈ B , and that if w ∈D ∆ T, B , then ( λI − ∆ T ) − (( λI − ∆ T ) w ) = w. The situation is slightly more com-plicated if we consider ∆ T as a quasimultiplier on A when lim sup t → + k T ( t ) k =+ ∞ . In this case the domain D ( λI − ∆ T ) − , A of ( λI − ∆ T ) − ∈ Q M ( A ) is a propersubspace of A containing L T ⊃ ∪ t> T ( t ) A , and we have ( λI − ∆ T ) − v ∈ D ∆ T, A and ( λI − ∆ T ) (cid:0) ( λI − ∆ T ) − v (cid:1) = v for every v ∈ D ( λI − ∆ T ) − , A . Also if w ∈D ∆ T , A , then ( λI − ∆ T ) w ∈ D ( λI − ∆ T ) − , A , and we have ( λI − ∆ T ) − (( λI − ∆ T ) w ) = w. In order to interpret ( λI − ∆ T ) − as a partially defined operator on A for Re ( λ ) > log ( ρ T ) , we can use the formula ( λI − ∆ T ) − v = Z + ∞ e − λt T ( t ) vdt ( v ∈ L T ) , (9)which defines a quasimultiplier on A if we apply it to some v ∈ Ω( A ) suchthat lim sup t → + k T ( t ) u k < + ∞ . The fact that this quasimultiplier is regu-lar is not completely obvious but follows from the previous discussion since ( λI − ∆ T ) − ∈ M ( B ) ⊂ Q M r ( A ) . Notice that since ∪ t> T ( t ) A is dense in ( U T , k . k U T ) , ( λI − ∆ T ) − is characterized by the simpler formula ( λI − ∆ T ) − T ( s ) v = e λs Z + ∞ s e − λt T ( t ) vdt ( s > , v ∈ A ) . (10) Let a < b ≤ a + π. In this section we consider a holomorphic semigroup T =( T ( ζ )) ζ ∈ S a,b of multipliers on a weakly cancellative commutative Banach algebra A having dense principal ideals such that T ( ζ ) A is dense in A for some, or,equivalently, for every ζ ∈ S a,b . Denote by I T the closed span of { T ( ζ ) } ζ ∈ S a,b , which is equal to the closedspan of { T ζ ( t ) } t> for ζ ∈ S a,b . For ζ ∈ S a,b , set T ζ = ( T ( tζ )) t> , let Φ T ζ : ω Tζ → M ( A ) be the homomorphism defined by (2). Set ω T ( ζ ) = k T ( ζ ) k for ζ ∈ S a,b , denote by M ω T ( S a,b ) the space of all measures µ on S a,b such that k µ k ω T := R S a,b ω T ( ζ ) d | µ | ( ζ ) < + ∞ , which is a Banach algebra with respect toconvolution. The convolution algebra L ω T ( S a,b ) is defined in a similar way andwill be identified to the closed ideal of M ω T consisting of measures which areabsolutely continuous with respect to Lebesgue measure. Define Φ T : M ω T →I T ⊂ M ( A ) by the formula Φ T ( µ ) = Z S a,b T ( ζ ) dµ ( ζ ) , which is well-defined since the map ζ → T ( ζ ) is continuous with respect tothe norm topology on M ( A ) and since I T is separable.Let ζ ∈ S a,b . Since the semigroup T ζ is continuous with respect to the normtopology on M ( A ) , a standard argument shows that we have, for every Diracsequence ( f n ) n ≥ , lim sup n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ ( f n ∗ δ s )( t ) T ( tζ ) dt − T ( sζ ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 , and so T ( sζ ) ∈ I T ζ for every s > , which implies that I T ζ = I T , and a similarargument shows that I T is the closure in ( M ( A ) , k . k M ( A ) ) of Φ T ( M ω T ( S a,b )) , aswell as the closure of Φ T ( L ω T ( S a,b )) and the closure of Φ T ζ ( L ω Tζ ) in ( M ( A ) , k . k M ( A ) ) ,and the notation I T is consistent with the notation used to denote the Arve-son ideal associated to a strongly continuous semigroup of multipliers on thehalf-line.The following interpretation of the generator of a holomorphic semigroupas a quasimultiplier follows the interpretation given in [7] in the case where A = I T . Proposition 7.1
Set ∆ T, A := S T ′ ( ζ ) u /T ( ζ ) u ∈ QM ( A ) , where ζ ∈ S a,b , u ∈ Ω( A ) . Then this definition does not depend on the choice of ζ and u , and we have,for ζ ∈ S a,b , ∆ T ζ , A = ζ ∆ T, A , (11) where the generator ∆ T ζ , A of the semigroup T ζ is the quasimultiplier on A introduced in definition 4.1.Moreover If T = ( T ( ζ )) ζ ∈ S a,b and T = ( T ( ζ )) ζ ∈ S a,b are two holomorphicsemigroups of multipliers on A such that T ( ζ ) A and T ( ζ ) A are dense in A and such that T ( ζ ) T ( ζ ) = T ( ζ ) T ( ζ ) for ζ ∈ S α,β , then we have ∆ T T , A = ∆ T , A + ∆ T , A . ζ ∈ S a,b ,T ′ ( ζ ) T ( ζ ) = T ′ ( ζ + ζ ) = T ′ ( ζ ) T ( ζ ) , and so the definition of ∆ T, A does not depend on the choice of ζ , and aneasy argument given in the comments following definition 4.1 shows that thisdefinition does not depend on the choice of u ∈ Ω( A ) either.Now let ζ ∈ S a,b , and le f ∈ C ([0 , + ∞ )) ∩ Ω( L ω Tζ ) such that R + ∞ | f ( t ) |k T ( tζ ) k dt < ∞ and R + ∞ | f ′ ( t ) |k T ( tζ ) k dt < ∞ satisfying f (0) = 0 . We have, integrating byparts, since lim p → + ∞ | f ( n p ) |k T ( n p ζ ) k = 0 for some strictly increasing sequence ( n p ) p ≥ of integers, T ( ζ ) Z + ∞ f ′ ( t ) T ( tζ ) dt = lim p → + ∞ Z n p f ′ ( t ) T ( ζ + tζ ) dt = lim p → + ∞ (cid:18) [ f ( t ) T ( ζ + tζ )] n p − ζ Z n p f ( t ) T ′ ( ζ + tζ ) dt (cid:19) = − ζT ′ ( ζ ) Z + ∞ f ( t ) T ( tζ ) dt, and formula (11) follows since (cid:16)R + ∞ f ( t ) T ( tζ ) dt (cid:17) u = φ T ζ ( f ) u ∈ Ω( A ) for u ∈ Ω( A ) . The last assertion follows immediately from the Leibnitz rule . (cid:3)
The following corollary follows then from proposition 5.4.
Corollary 7.2 (i) Let u ∈ A , and let ζ ∈ S a,b . If lim t → + (cid:13)(cid:13)(cid:13) T ( tζ ) u − ut − v (cid:13)(cid:13)(cid:13) = 0 for some v ∈ A , then u ∈ D ∆ T, A , and ζ ∆ T, A u = v. (ii) Conversely if lim sup t → + k T ( tζ ) k < + ∞ , then lim sup t → + (cid:13)(cid:13)(cid:13) T ( t ) u − ut − ζ ∆ T, A u (cid:13)(cid:13)(cid:13) =0 for every u ∈ D ∆ T, A . In the remaining of the section we will denote by B a normalization of A with respect to the semigroup T, see definition 4.3. Since QM ( A ) is isomorphicto QM ( B ) , we can consider the generator ∆ T, A as a quasimultiplier on B , andit follows immediately from definition 7.1 that this quasimultiplier on B is thegenerator of the semigroup T considered as a semigroup of multipliers on B . From now on we will thus set ∆ T = ∆ T, A = ∆ T, B . Applying corollary 7.2 to T and B , we obtain Corollary 7.3 (i) Let u ∈ B Then the following conditions imply each other(i) There exists ζ ∈ S a,b and v ∈ B such that lim t → + (cid:13)(cid:13)(cid:13) T ( ζ t ) u − ut − v (cid:13)(cid:13)(cid:13) B = 0 , (ii) u ∈ D ∆ T , B , and in this situation lim t → + (cid:13)(cid:13)(cid:13) T ( ζt ) u − ut − ζ ∆ T u (cid:13)(cid:13)(cid:13) B = 0 for every ζ ∈ S a,b . Denote by c I T the space of characters on I T , equipped with the usual Gelfandtopology. If χ ∈ c I T , the map ζ → χ ( T ( ζ )) is holomorphic on S a,b , and so there28xists a unique complex number c χ such that χ ( T ( ζ )) = e ζc χ for ζ ∈ S a,b . Wesee as in section 5 that there exists a unique character ˜ χ on QM ( I T ) suchthat ˜ χ | I T = χ, and since ∆ T ζ , I T = ζ ∆ T, I T it follows from proposition 5.7 andproposition 7.1 that χ ( T ( tζ )) = e t ˜ χ (∆ Tζ, I T ) = e tζ ˜ χ (∆ T, I T ) for ζ ∈ S a,b , t > , and so c χ = ˜ χ (∆ T, I T ) . Since ∆ T ζ , I T = ζ ∆ T, I T for ζ ∈ S a,b , we deduce from corollary 5.8 andproposition 6.1 the following result. Proposition 7.4
Let T = ( T ( ζ ) ζ ∈ S a,b ⊂ M ( A ) be a holomorphic semigroup.Set σ ar (∆ T, I T ) = { ˜ χ (∆ T, I T ) } χ ∈ c I T , with the convention σ ar (∆ T, I T ) = ∅ ifthe semigroup is quasinilpotent, and set Res ar (∆ T ) = Res ar (∆ T, I T ) = C \ σ ar (∆ T, I T ) . Let B be a normalization of A with respect to the holomorphicsemigroup T, and let I T, B be the closed subalgebra of M ( B ) generated by thesemigroup.(i) The set Λ t,ζ := { λ ∈ σ ar (∆ T , I T ) | Re ( λζ ) ≤ t } is compact for ζ ∈ S a,b , t ∈ R . (ii) The quasimultiplier λI − ∆ T has an inverse ( λI − ∆ T ) − ∈ I T, B ⊂M ( B ) ⊂ QM r ( A ) for λ ∈ Res ar (∆ T ) , and the map λ → ( λI − ∆ T ) − is aholomorphic map from Res ar (∆ T ) into I T, B . (iii) If ζ ∈ S a,b , then λ ∈ Res ar (∆ T ) for Re ( λζ ) > lim t → + ∞ log ( k T ( tζ ) k ) t , and we have ( λI − ∆ T ) − = Z ζ. ∞ e − sλ T ( s ) ds, (12) so that k ( λI − ∆ T ) − k M ( B ) ≤ | ζ | Z + ∞ e − tRe ( λζ ) k T ( tζ ) k M ( B ) dt. (13)Proof: (i) Let ζ ∈ S a,b , t > , and set V = { λ ∈ ζσ ar (∆ T, B T ) | Re ( λ ) ≤ t } = { λ ∈ σ ar (∆ T ζ , I T ) | Re ( λ ) ≤ t } . It follows from corollary 4.7 that V is compact,and so Λ t,ζ = ζ − V is compact.(ii) Fix ζ ∈ S a,b . We have λI − ∆ T = λI − ζ − ∆ T ζ = ζ − (cid:0) λζ I − ∆ T ζ (cid:1) . If λ ∈ Res ar (∆ T ) , then λI − ∆ T is invertible in QM ( A ) , and ( λI − T ) − = ζ − ( λζ I − ∆ T ζ ) − ∈ I T, B ⊂ M ( B ) ⊂ QM r ( A ) , since in this situation λζ ∈ Res (∆ T ζ ) , and it follows also from proposition 6.1 that the I T, B -valued map λ → ( λI − T ) − = ζ − ( λζ I − ∆ T ζ ) − is holomorphic on Res ar (∆ T ) . (iii) This follows from proposition 6.1 applied to λζ and T ζ . (cid:3) Multivariable functional calculus for holomor-phic semigroups associated to linear function-als
In the following definition, we write by convention T j (0) = I for ≤ j ≤ k. Set σζ = σ ζ + . . . + σ k ζ k for σ = ( σ , . . . , σ k ) , ζ = ( ζ , . . . , ζ k ) ∈ C k . Let a = ( a , . . . , a k ) ∈ R k , b = ( b , . . . , b k ) ∈ R k such that a j ≤ b j ≤ a j + π for ≤ j ≤ k. As in appendix 2, we set M a,b = { ( α, β ) ∈ R k × R k | a j < α j ≤ β j < b j if a j < b j , α j = β j = a j if a j = b j } . Definition 8.1 : Let a = ( a , . . . , a p ) ∈ R k , b = ( b , . . . , b p ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k, let A be a weakly cancellative commutative Banachalgebra with dense principal ideals, and let T = ( T , . . . , T k ) be a family ofsemigroups of multipliers on A which possesses the following properties T j = ( T j ( ζ )) ζ ∈ (0 ,a j . ∞ ) is strongly continuous on (0 , e ia j . ∞ ) , and ∪ t> T ( te ia j ) A is dense in A if a j = b j ,T j = ( T ( ζ )) ζ ∈ S aj,bj is holomorphic on S a j ,b j , and T ( ζ ) A is dense in A for every ζ ∈ S a j ,b j if a j < b j . For ζ = ( ζ , . . . , ζ k ) ∈ ∪ ( α,β ) ∈ M a,b S α,β set T ( ζ ) = T ( ζ ) . . . T k ( ζ k ) . A subalgebra B of QM ( A ) is said to be a normalization of A with respect to T if the following conditions are satisfied(a) ( B , k . k B ) is a Banach algebra with respect to a norm k . k B satisfying k u k B ≤ k u k A for u ∈ A , and there exists a family ( w , . . . , w k ) of elementsof Ω( A ) such that the inclusion map j : A → B is a QM -homomorphismwith respect to T ( ζ ) . . . T k ( ζ k ) w . . . w k for every family ( ζ , . . . , ζ k ) of complexnumbers such that ζ j ∈ S a j ,b j if a j < b j and such that ζ j = 0 , if a j = b j . (b) ˜ j ( M ( A )) ⊂ M ( B )) , and k ˜ j ( R ) k M ( B ) ≤ k R k M ( A ) for every R ∈ M ( A ) , where ˜ j : QM ( A ) → QM ( B ) is the pseudobounded isomorphism associated to j in proposition 2.2 (ii).(c) lim sup ζ → ζ ∈ Sγ,δ k T ( ζ ) k M ( B ) < + ∞ for a j < γ < δ < b j if a j < b j , and lim sup t → + k T ( te ia j ) k M ( B ) < + ∞ if a j = b j . It follows from proposition 3.2 and proposition 4.2 that there exists a nor-malization B of A with respect to T . Also if B m is a normalization of A withrespect to ( T , . . . , T m ) and if B m +1 is a normalization of B m with respect to T m +1 , it follows from proposition 2.4 and definitions 3.3 and 4.3 that B m +1 isa normalization of A with respect to ( T , . . . , T m +1 ) . It is thus immediate toconstruct a normalization of A with respect to T by a finite induction. Noticethat if B is a normalization of A with respect to T, then B is a normalizationof A with respect to T σ := ( T ( tσ )) t> for every σ ∈ ∪ ( α,β ) ∈ M a,b S α,β .30ince ∪ t> T ( te ia j ) B is dense in B , when a j = b j , and since T ( ζ ) A is dense in A for ζ ∈ S a j ,b j if a j < b j , it follows from condition (c) of definition 10.1 that themap ζ → T ( ζ ) u . . . u k is continuous on S α,β for ( α, β ) ∈ M a,b , u , . . . , u k ∈ B . Since u . . . u k ∈ Ω( B ) for u , . . . , u k ∈ Ω( B ) , it follows again from condition (c)of definition 10.1 that the map ζ → T ( ζ ) u is continuous on S α,β for ( α, β ) ∈ M a,b for every u ∈ B . Let ( α, β ) ∈ M a,b and assume that a j < b j . Since the semi-group T j is holomorphic on S a j ,b j the map η → T ( ζ , ζ , . . . , ζ j − , η, ζ j +1 , ζ k ) u is holomorphic on S α j ,β j for every ( ζ , . . . , ζ j − , ζ j +1 , ζ k ) ∈ Π ≤ s ≤ ks = j S α s ,β s . Notice that if u ∈ B , where B is a normalization of A with respect to T, thenthe closed subspace B T,u spanned by the set { T ( ζ ) u | ζ ∈ S a,b } is separable,and so the function ζ → T ( ζ ) u takes its values in a closed separable subspaceof B . With the convention T j (0) = I for ≤ j ≤ k, we see that if ( α, β ) ∈ M a,b andif λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ then T ( λ ) : ζ → T ( λζ ) = ( T ( λ ζ ) , . . . , T k ( λ k ζ k )) iswell-defined for ζ ∈ S α,β . Proposition 8.2
Let ( α, β ) ∈ M a,b . For λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ , denote by N ( T, λ, α, β ) the set of all z ∈ C k such that lim sup t → + ∞ | e tz j e iω |k T j ( tλ j e iω ) k < + ∞ for α j ≤ ω ≤ β j , ≤ j ≤ k , and denote by N ( T, λ, α, β ) the set of all z ∈ C k such that lim t → + ∞ | e tz j e iω |k T j ( tλ j e iω ) k = 0 for α j ≤ ω ≤ β j , ≤ j ≤ k. Then z ∈ N ( T, λ, α, β ) if and only if lim sup t → + ∞ | e tz j e iαj |k T j ( tλ j e iα j ) k < + ∞ and lim sup t → + ∞ | e tz j e iβj |k T j ( tλ j e iβ j ) k < + ∞ for ≤ j ≤ k, Also z ∈ N ( T, λ, α, β ) if and only if Re ( z j e iα j ) < − lim t → + ∞ log k T ( tλ j e iαj ) k t and Re ( z j e iβ j ) < − lim t → + ∞ log k T ( tλ j e iβj ) k t for ≤ j ≤ k. Proof: Let j ≤ k such that α j < β j . If α j ≤ ω ≤ β j , there exists r > and s > such that e iω = r e iα j + s e iβ j , and we have, for z j ∈ C , | e tz j e iω |k T j ( tλ j e iω k ≤ | e r tz j e iαj |k T j ( r tλ j e iα j k| e s tz j e iβj |k T j ( s tλ j e iβ j k , (14)and we see that z ∈ N ( T, λ, α, β ) if and only if lim sup t → + ∞ | e tz j e iαj |k T j ( tλ j e iα j ) k < + ∞ and lim sup t → + ∞ | e tz j e iβj |k T j ( tλ j e iβ j ) k < + ∞ for ≤ j ≤ k, which impliesthat Re ( z j e iα j ) ≤ − lim t → + ∞ log k T ( tλ j e iαj k t and Re ( z j e iβ j ) ≤ − lim t → + ∞ log k T ( tλ j e iβj k t for ≤ j ≤ k. A similar argument shows that z ∈ N ( T, λ, α, β ) if and only if Re ( z j e iα j ) < − lim t → + ∞ log k T ( tλ j e iαj k t and Re ( z j e iβ j ) < − lim t → + ∞ log k T ( tλ j e iβj k t ,which implies that Re ( z j e iω ) < − lim t → + ∞ log k T ( tλ j e iω k t for α j ≤ ω ≤ β j , ≤ j ≤ k. (cid:3) N ( T, λ, α, β ) − S ∗ α,β ⊂ N ( T, λ, α, β ) and that N ( T, λ, α, β ) − S ∗ α,β ⊂ N ( T, λ, α, β ) . Set again e z ( ζ ) = e zζ for z ∈ C k , ζ ∈ C k . If B is a normalization of A with respect to T, then sup | ζ |≤ ζ ∈ S α,β k T ( ζ ) k M ( B ) < + ∞ for ( α, β ) ∈ M a,b , and it follows from (42) that sup ζ ∈ S α,β | e z ( ζ ) |k T ( λζ ) k M ( B ) < + ∞ for z ∈ N ( T, α, β, λ ) and lim | ζ |→ + ∞ ζ ∈ S α,β | e z ( ζ ) |k T ( λζ ) k M ( B ) = 0 if z ∈ N ( T, λ, α, β ) when λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − δ S γ,δ . With the notations of appendices 1 and 2, we obtainthe following result.
Proposition 8.3
Let ( α, β ) ∈ M a,b , and let λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ . (i) If z ∈ N ( T, λ, α, β ) , then e z T ( λ. ) u | Sα,β ∈ V α,β ( B ) , ζ j − z j ∈ Res ar ( λ j ∆ T j ) for ζ ∈ S ∗ a,b , u ∈ B , ≤ j ≤ k, and we have FB ( e z T ( λ. ) u | Sα,β )( ζ ) = ( − k (( z − ζ ) I + λ ∆ T ) − . . . (( z k − ζ k ) I + λ k ∆ T k ) − u. (ii)If z ∈ N ( T, λ, α, β ) then e z T ( λ. ) u | Sα,β ∈ U α,β ( B ) , FB ( e z T ( λ. ) u | Sα,β hasa continuous extension to S ∗ α,β , z j + ζ j ∈ Res ar ( λ j ∆ j ) for ≤ j ≤ k, and wehave, for ζ ∈ S ∗ a,b , u ∈ B , FB ( e z T ( λ. ) u | Sα,β )( ζ ) = ( − k (( z − ζ ) I + λ ∆ T ) − . . . (( z k − ζ k ) I + λ k ∆ T k ) − u. Proof: It follows from the discussion above that e z T ( λ. ) u | Sα,β ∈ V α,β ( B ) if z ∈ N ( T, λ, α, β ) , and that e z T ( λ. ) u | Sα,β ∈ U α,β ( B ) if z ∈ N ( T, λ, α, β ) . Let z ∈ N ( T, λ, α, β ) , and let u ∈ B . It follows from definition 10.3 (iii) that wehave, for ζ ∈ S ∗ a,b , FB ( e z T ( λ. ) u | Sα,β )( ζ )= Z e iω . ∞ . . . Z e iωk . ∞ e ( z − ζ ) σ + ... +( z k − ζ k ) σ k T ( λ σ ) . . . T k ( λ k σ k ) udσ . . . dσ k , where α j ≤ ω j ≤ β j and where Re ( ζ j e iω j ) > for ≤ j ≤ k. Since Re (( ζ j − z j ) e iω j ) > lim t → + ∞ log ( k T ( tλ j ω j k t , it follows from proposition6.1 and proposition 7.4 that ζ j − z j ∈ Res ar ( λ j ∆ T j ) for j ≤ k, and that wehave, for v ∈ B , Z e iωj . ∞ e ( z j − ζ j ) σ j T ( λ j σ j ) vdσ j = − (( z j − ζ j ) I + λ j ∆ T j ) − v. Using Fubini’s theorem, we obtain FB ( e z T ( λ. ) u | Sα,β )( ζ ) − (( z − ζ ) I + λ ∆ T ) − Z e iω . ∞ . . . Z e iωk . ∞ e ( z − ζ ) σ − ... +( z k − ζ k ) σ k ) T ( λ σ ) . . . T k ( λ k σ k ) udσ . . . dσ k = . . . = ( − k (( z − ζ ) I + λ ∆ T ) − . . . (( z k − ζ k ) I + λ k ∆ T k ) − u. Since N ( T, λ, α, β ) ⊂ N ( T, λ, α, β ) − S ∗ α,β , (ii) follows then from (i). (cid:3) Recall that F α,β = ( ∩ z ∈ C k e − z U α,β ) ′ = ∪ z ∈ C k ( e − z U α,β ) ′ . If φ ∈ F α,β , then Dom ( FB ( φ )) is the set of all z ∈ C k such that φ ∈ ( e − z U α,β ) ′ . In the followingdefinition the action of φ ∈ F α,β on an element f of e − z V α,β ( B ) taking values ina closed separable subspace of B , where z ∈ Dom ( FB ( φ ) , is defined accordingto definition 11.3. by the formula < f, φ > = < e z f, φe − z >, where < g, φe − z > = < e − z g, φ > for g ∈ U α,β ( B ) . It follows from the remarksfollowing definition 11.3 that the above definition does not depend on the choiceof z. Definition 8.4
Let ( α, β ) ∈ M a,b , let λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ , let φ ∈ F α,β , and let B be a normalization of A with respect to T. For ζ ∈ S α,β , set T ( λ ) ( ζ ) = T ( λ ζ , . . . , λ k ζ k ) = T ( λ ζ ) . . . T k ( λ k ζ k ) , with the convention T j (0) = I. If N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , set, for u ∈ B ,< T ( λ ) , φ > u = < T ( λ. ) u | Sα,β , φ > .
For ( α, β ) ∈ M a,b , z (1) ∈ C k , z (2) ∈ C k , we define as in definition 11.1 sup( z (1) , z (2) ) to be the set of all z ∈ C k such that z + S ∗ α,β = ( z (1) + S ∗ α,β ) ∩ ( z (2) + S ∗ α,β ) , so that sup( z (1) , z (2) ) is a singleton if a j < b j for j ≤ k. Lemma 8.5 If φ ∈ F α,β , φ ∈ F α,β , and if N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , then sup( z (1) , ( z (2) ) ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) ∩ Dom ( FB ( φ )) ⊂ N ( T, λ, α, β ) ∩ Dom ( FB ( φ ∗ φ )) for z (1) ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , z (2) ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , and the sameproperty holds for N ( T, λ, α, β ) . Proof: Let z (1) ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , let z (2) ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , let z ∈ sup( z (1) , z (2) ) , and let j ≤ k. There exists s ∈ { , } and s ∈ { , } such that z j ∈ (cid:16) z ( s ) j + [0 , e ( − π − α j ) i . ∞ ) (cid:17) ∩ (cid:16) [ z ( s ) j + [0 , e ( π − β j ) i . ∞ ) (cid:17) , and it follows from (17) and from proposition 8.2 that z ∈ N ( T, λ, α, β ) . Thefact that z ∈ Dom ( φ ∗ φ ) follows from proposition 11.6. A similar argumentshows that the same property holds for N ( T, λ, α, β ) . (cid:3) Theorem 8.6
Let A be a weakly cancellative commutative Banach algebra withdense principal ideals, let a, b ∈ R k , let T = ( T , . . . , T k ) be a family of semi-groups of multipliers on A satisfying the conditions of definition 8.1, let B be a normalization of A with respect to T, let ( α, β ) ∈ M a,b and let λ ∈∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ . f N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ for some φ ∈ F α,β , then the followingproperties hold(i) < T ( λ ) , φ > ∈ M ( B ) ⊂ QM r ( A ) , and we have, for z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , if ν is a z -representative measure for φ,< T ( λ ) , φ > = Z S α,β e zζ T ( λζ ) dν ( ζ ) , where the Bochner integral is computed with respect to the strong operatortopology on M ( B ) , and if χ is a character on A , then we have ˜ χ (cid:0) < T ( λ ) , φ > (cid:1) = FB ( φ )( − λ ˜ χ (∆ T ) , . . . , − ˜ χ (∆ T k )) , where ˜ χ denotes the unique character on QM ( A ) such that ˜ χ | A = χ. (ii) lim η → (0 ,..., ,η ∈ Sα,βǫ → (0 ,..., ,ǫ ∈ S ∗ α,β k < e − ǫ T ( λ ) , φ ∗ δ η > u − < T ( λ ) , φ > u k B = 0 for u ∈ B . (iii) If α j < β j < α j + π for ≤ j ≤ k, then we have, for η ∈ S α,β , ǫ ∈ S ∗ α,β ,< e − ǫ T ( λ ) , φ ∗ δ η > = e − zη Z ˜ ∂S α,β e ( z − ǫ ) σ C z ( φ )( σ − η ) T ( λσ ) dσ where the Bochner integral is computed with respect to the strong operatortopology on M ( B ) . (iv) In the general case, set W n ( ζ ) = Π ≤ j ≤ k n n + ζ j e i αj + βj ! for n ≥ , ζ =( ζ , . . . , ζ n ) ∈ S ∗ αβ . Then we have < T ( λ ) , φ > = lim ǫ → ǫ ∈ S ∗ α,β lim n → + ∞ ( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β W n ( σ − z ) FB ( φ )( σ )(( σ − ǫ ) I + λ ∆ T ) − . . . (( σ k − ǫ k ) I + λ k ∆ T k ) − dσ ! , where the Bochner integral is computed with respect to the norm topology on M ( B ) . (v) If, further, R z +˜ ∂S ∗ α,β |FB ( φ )( σ ) || dσ ) | < + ∞ , then we have < T ( λ ) , φ > = lim ǫ → ǫ ∈ S ∗ α,β < e − ǫ T ( λ ) , φ > =lim ǫ → ǫ ∈ S ∗ α,β ( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ )(( σ − ǫ ) I + λ ∆ T ) − . . . (( σ k − ǫ k ) I + λ k ∆ T k ) − dσ, where the Bochner integral is computed with respect to the norm topology on M ( B ) . vi) If the condition of (v) is satisfied for some z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , then we have < T ( λ ) , φ > =( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ )(( σ I + λ ∆ T ) − . . . ( σ k I + λ k ∆ T k ) − dσ. (vii) If φ ∈ F α,β , φ ∈ F α,β , and if N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ and N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , then N ( T, λ, α, β ) ∩ Dom ( FB ( φ ∗ φ )) = ∅ , and < T ( λ ) , φ ∗ φ > = < T ( λ ) , φ >< T ( λ ) , φ > . Proof: (i) Let z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , and set m = sup ζ ∈ S α,β | e zζ |k T λ ( ζ ) k M ( B ) < + ∞ . We have k < T λ , φ > u k B ≤ m k φe z k U ′ α,β k u k B , and so < T λ , φ > ∈ M ( B ) . The integral formula in (i) follows then immedi-ately from the definition given in proposition 10.2 and from definition 11.3.Assume that A is not radical, let χ be a character on A , and let ˜ χ bethe unique character on QM ( A ) such that ˜ χ ( u ) = χ ( u ) for every u ∈ A . Set f n ( t ) = 0 if ≤ t < n +1 or if t > n , and f n ( t ) = n ( n + 1) if n +1 ≤ t ≤ n , andlet ζ be an element of the domain of definition of T j . Set T j,ζ := ( T j ( tζ )) t> . Then ( f n ) n ≥ ⊂ L ω Tj,ζ ( R + ) is a Dirac sequence, and since the map t → T j ( tζ ) u is continuous on (0 , + ∞ ) , a standard argument shows that we have, for s > ,u ∈ A , lim sup n → + ∞ (cid:13)(cid:13) Φ T j,ζ ( f n ) T j ( sζ ) u − T j ( sζ ) u (cid:13)(cid:13) = lim sup n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ ( f n ∗ δ s )( t ) T j ( tζ ) udt − T j ( sζ ) u (cid:13)(cid:13)(cid:13)(cid:13) = 0 . Since ∪ t> T j ( tζ )( A ) is dense in A , there exist n ≥ such that ˜ χ (Φ T j,ζ ( f n )) =0 , and the restriction of ˜ χ to the Arveson ideal I T j,ζ is a character on I T j,ζ . It fol-lows then from proposition 5.7 and proposition 7.1 that ˜ χ ( T j ( tζ )) = e t ˜ χ (∆ Tj,ζ ) = e tζ ˜ χ (∆ Tj ) for t > , and so ˜ χ ( T j ( ζ )) = e ζ ˜ χ (∆ Tj ) for every ζ in the domain of defi-nition of T j . Let u ∈ M ( B ) . By continuity, we see that ˜ χ ( T j ( ζ ) u ) = e ζ ˜ χ (∆ Tj ) ˜ χ ( u ) for every ζ ∈ S α j ,β j . Set λζ ˜ χ (∆ T ) = λ ζ ˜ χ (∆ T ) + . . . + λ k ζ k ˜ χ (∆ T k ) . Consideragain z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) . Since FB ( φe − z )( ζ ) = < e − ζ , φe − z > =
In the following definition, the generator ∆ T j of the strongly continuous semi-group T j and its Arveson spectrum σ ar (∆ T j ) are defined according to section5 if a j = b j , and the generator ∆ T j of the holomorphic semigroup T j and itsArveson spectrum σ ar (∆ T j ) are defined according to section 7 if a j < b j . Definition 9.1
Let a = ( a , . . . , a p ) ∈ R k , b = ( b , . . . , b p ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k, let A be a weakly cancellative commutative Banachalgebra having dense principal ideals, and let T = ( T , . . . , T k ) be a family ofsemigroups of multipliers on A satisfying the conditions of definition 8.1. Let ( α, β ) ∈ M a,b and let λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ . An open set U ⊂ C k is said to be admissible with respect to ( T, λ, α, β ) if U = Π ≤ j ≤ k U j where the open sets U j ⊂ C satisfy the following conditions forsome z = ( z , . . . , z k ) ∈ N ( T, α, β, λ ) (i) U j + S ∗ α j ,β j ⊂ U j (ii) U j ⊂ z j + S ∗ α j ,β j , and ∂U j = ( z j + e ( − π − α j ) i . ∞ , z j + e ( − α j − π ) i s ,j ) ∪ γ ([0 , ∪ ( z j + e ( π − β j ) i s ,j , z j + e ( π − β j ) i . ∞ ) , where s ,j ≥ , s ,j ≥ , and where γ : [0 , → z j + S ∗ α j ,β j \ (cid:0) e ( − π − α j ) i . ∞ , e ( − α j − π ) i s ,j ) ∪ ( e ( π − β j ) i s ,j , e ( π − β j ) i . ∞ ) (cid:1) is a one-to-one piecewise- C curve such that γ (0) = e ( − α j − π ) i s ,j and γ = e ( π − β j ) i s ,j . (iii) λ j σ ar ( − ∆ T j ) = σ ar ( − ∆ T j ( λ j . ) ) ⊂ U j . Conditions (i) and (ii) mean that U is admissible with respect to ( α, β ) in thesense of definition 12.1 and that some, hence all elements z ∈ C k with respect towhich U satisfies condition (ii) of definition 12.1 belong to N ( T, α, β, λ ) . Hence U j is a open half-plane if α j = β j , and the geometric considerations about ∂U j made in section 12 when α j < β j apply.37or α = ( α , . . . , α k ) ∈ R k , β = ( β , . . . , β k ) ∈ R k , we will use as in appendix3 the obvious conventions inf( α, β ) = (inf( α , β ) , . . . , inf( α k , β k )) , sup( α, β ) = (sup( α , β ) , . . . , sup( α k , β k )) . Proposition 9.2 If U (1) is admissible with respect to ( T, λ, α (1) , β (1) ) and if U (2) is admissible with respect to ( T, λ, α (2) , β (2) ) , then U (1) ∩ U (2) is admissiblewith respect to ( T, λ, inf( α (1) , α (2) ) , sup( β (1) , β (2) )) . Proof: Set α (3) = inf( α (1) , α (2) ) , β (3) = sup( β (1) , β (2) ) , and set U (3) = U (1) ∩ U (2) . Then h ∪ ( γ,δ ) ∈ M a − α (1) ,b − β (1) S γ,δ i ∩ h ∪ ( γ,δ ) ∈ M a − α (2) ,b − β (2) S γ,δ i ⊂ h ∪ ( γ,δ ) ∈ M a − α (3) ,b − β (3) S γ,δ i , so it makes sense to check whether U (1) ∩ U (2) is admissible with respect to ( T, λ, α (3) , β (3) ) . The fact that U (3) satisfies (i) and (ii) follows from proposition12.2, and the fact that U (3) satisfies (iii) is obvious. (cid:3) If an open set U ⊂ C k is admissible with respect to ( T, λ, α, β ) , we denote asin section 12 by H (1) ( U ) the set of all holomorphic functions F on U satisfyingthe condition k F k H (1) ( U ) := sup ǫ ∈ S ∗ α,β Z σ ∈ ˜ ∂U + ǫ | F ( σ ) || dσ | < + ∞ . Notice that ∪ ( α,β ) ∈ M a,b (cid:0) ∪ ( γ,δ ) ∈ M a − α,b − β S β,γ (cid:1) = ∪ ( α,β ) ∈ M a,b S a − α,b − β . Theinclusion ∪ ( γ,δ ) ∈ M a − α,b − β S β,γ ⊂ S a − α,b − β for ( α, β ) ∈ M a,b is obvious. Con-versely assume that λ ∈ S a − α,b − β for some ( α, β ) ∈ M a,b . If a j = b j then a j = α j = β j = b j , and so λ j is a nonnegative real number. In this situationset α ′ j = β ′ j = a j , γ j = δ j = 0 . If a j < b j , then a j < α j ≤ β j < b j , and a j − α j ≤ arg ( λ j ) ≤ b j − β j if λ j = 0 . In this situation set α ′ j = a j + α j , γ j = a j − α j , δ j = b j − β j and β ′ j = b j + β j . Then ( α ′ , β ′ ) ∈ M a,b , ( γ, δ ) ∈ M a − α ′ ,b − β ′ , and λ ∈ S γ,δ , which concludes the proof of the reverse inclusion. Corollary 9.3
For λ ∈ ∪ ( α,β ) ∈ M a,b S a − α,b − β = ∪ α,β ∈ M a,b (cid:0) ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ (cid:1) , denote by N λ the set of all ( α, β ) ∈ M a,b such that λ ∈ ∪ ( γ,δ ) ∈ M a − α,b − β S γ,δ , anddenote by W T,λ the set of all open sets U ⊂ C k which are admissible with respectto ( T, λ, α, β ) for some ( α, β ) ∈ N λ . Then W T,λ is stable under finite intersections, and ∪ U ∈W T,λ H (1) ( U ) is stableunder products. Proof: The first assertion follows from the proposition and the second as-sertion follows from the fact that the restriction of F ∈ H (1) ( U ) is boundedon U + ǫ if U is admissible with respect to ( α, β ) ∈ M a,b and if ǫ ∈ S ∗ α,β , seecorollary 12.4. (cid:3) A set
E ⊂ ∪ U ∈W T,λ H (1) ( U ) will be said to be bounded if there exists U ∈ W T,λ such that
E ⊂ H (1) ( U ) and such that sup F ∈E k F k H (1) ( U ) < + ∞ , and bounded subsets of ∪ U ∈W T,λ H ∞ ( U ) are defined in a similar way. A homo-morphism φ : ∪ U ∈W T,λ H (1) ( U ) → M ( B ) will be said to be bounded if φ ( E ) is38ounded for every bounded subset E of ∪ U ∈W T,λ H (1) ( U ) , and a homomorphism φ : ∪ U ∈W T,λ H ∞ ( U ) → QM r ( B ) = QM r ( A ) will be said to be bounded if φ ( E ) is pseudobounded for every bounded subset E of ∪ U ∈W T,λ H ∞ ( U ) . Similarly let S ( U ) be the Smirnov class on U ∈ W T,λ introduced in definition12.6. A set
E ⊂ ∪ U ∈W T,λ S ( U ) will be said to be bounded if there exists U ∈W T,λ such that
E ⊂ S ( U ) and such that sup F ∈E k F G k H ∞ ( U ) < + ∞ for somestrongly outer function G ∈ H ∞ ( U ) , and a homomorphism φ : ∪ U ∈W T,λ S ( U ) →QM ( B ) = QM ( A ) will be said to be bounded if φ ( E ) is pseudobounded forevery bounded subset E of ∪ U ∈W T,λ S ( U ) . Let U = Π j ≤ k U j ∈ W T,λ , and let ( α, β ) ∈ N λ such that U is admissible withrespect to ( T, λ, α, β ) . Let ∂U j be oriented from e − π − α j . ∞ to e π − β j . ∞ . Thisgives an orientation on the distinguished boundary ˜ ∂U = Π j ≤ k ∂U j of U, to beused in the following theorem. Theorem 9.4
Let a = ( a , . . . , a p ) ∈ R k , b = ( b , . . . , b p ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k, let A be a weakly cancellative commutative Banach algebrawith dense principal ideals, let T = ( T , . . . , T k ) be a family of semigroups ofmultipliers on A satisfying the conditions of definition 8.1 with respect to ( a, b ) and A and let B be a normalization of A with respect to T. (i) For λ ∈ ∪ ( α,β ) ∈ M a,b S α,β , U ∈ W T,λ , F ∈ H (1) ( U ) , set F ( − λ ∆ T , . . . , − λ k ∆ T k )= 1(2 iπ ) k Z ˜ ∂U + ǫ F ( ζ , . . . , ζ k )( λ ∆ T + ζ I ) − . . . ( λ ∆ T k + ζ k I ) − dζ . . . dζ k , where U is admissible to respect to ( T, λ, α, β ) , with ( α, β ) ∈ N λ , and where ǫ ∈ S ∗ α,β is such that U + ǫ ∈ W T,λ . Then this definition does not depend onthe choice of U and ǫ, and the map F → F ( − λ ∆ T , . . . , − λ k ∆ T k ) is a boundedalgebra homomorphism from ∪ U ∈W T,λ H (1) ( U ) into M ( B ) ⊂ QM r ( A ) . (ii) For every U ∈ W T,λ there exists G ∈ H (1) ( U ) ∩ H ∞ ( U ) such that G ( − λ ∆ T , . . . , − λ k ∆ T k )( B ) is dense in B , and for every F ∈ H ∞ ( U ) thereexists a unique R F ∈ QM r ( B ) = QM r ( A ) satisfying R F G ( − λ T , . . . , − λ k T k ) = ( F G )( − λ T , . . . , − λ k T k ) ( G ∈ H (1) ( U )) . The definition of R F does not depend on the choice of U, and if we set F ( − λ T , . . . , − λ k T k ) = R F the definition of F ( − λ ∆ T , . . . , − λ k ∆ T k ) agrees with the definition givenin (i) if F ∈ ∪ U ∈W T,λ H (1) ( U ) , and the map F → F ( − λ T , . . . , − λ k T k ) is abounded homomorphism from ∪ U ∈W T,λ H ∞ ( U ) into QM r ( B ) = QM r ( A ) . (iii) If ( α, β ) ∈ N λ , if φ ∈ F α,β , and if N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , then FB ( φ )( − λ ∆ T , . . . , − λ k ∆ T k ) = < T ( λ ) , φ > . In particular if F ( ζ ) = e − νζ j for some ν ∈ C such that νλ j ∈ ∪ ( γ j ,δ j ) ∈ M aj,bj S γ j ,δ j then F ( − λ ∆ T , . . . , − λ k ∆ T k = T j ( νλ j ) . iv) If ( α, β ) ∈ N λ , if φ ∈ F α,β , and if N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , then < T ( λ ) , φ > u = lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β FB ( φ )( − λ ∆ T + ǫ I, . . . , − λ k ∆ T k + ǫ k I ) u ( u ∈ B ) . (v) If U ∈ W T,λ , and if F ∈ H ∞ ( U ) is strongly outer on U, then there exists u ∈ Ω( B ) ∩ Dom ( F ( − λ ∆ T , . . . , − λ k ∆ T k )) such that F ( − λ ∆ T , . . . , − λ k ∆ T k ) u ∈ Ω( B ) . (vi) For every U ∈ W T,λ and every F ∈ S ( U ) there exists a unique R F ∈QM ( B ) = QM ( A ) satisfying R F G ( − λ T , . . . , − λ k T k ) = ( F G )( − λ T , . . . , − λ k T k ) for every G ∈ H ∞ ( U ) such that F G ∈ H ∞ ( U ) . The definition of R F does notdepend on the choice of U, and if we set F ( − λ T , . . . , − λ k T k ) = R F the def-inition of F ( − λ ∆ T , . . . , − λ k ∆ T k ) agrees with the definition given in (ii) if F ∈ ∪ U ∈W T,λ H ∞ ( U ) , the map F → F ( − λ T , . . . , − λ k T k ) is a bounded ho-momorphism from ∪ U ∈W T,λ S ( U ) into QM ( B ) = QM ( A ) , and we have, for χ ∈ b A , ˜ χ ( F ( − λ T , . . . , − λ k T k )) = F ( − λ ˜ χ (∆ T ) , . . . , − λ k ˜ χ (∆ T k )) ( F ∈ ∪ U ∈W T,λ S ( U )) , where ˜ χ is the character on QM ( A ) such that ˜ χ | A = χ. (vii) If F ( ζ , . . . , ζ k ) = − ζ j then F ( − λ ∆ , . . . , − λ k ∆ k ) = λ j ∆ T j . Proof: In the following we will use the notations dζ = dζ . . . dζ k , λ ∆ T =( λ ∆ T , . . . , λ k ∆ T k ) , R ( − λ ∆ T , ζ ) = ( − k ( λ ∆ T + ζ I ) − . . . ( λ k ∆ T k + ζ k I ) − ) for ζ = ( ζ , . . . , ζ k ) ∈ − Res ar ( λ ∆ T ) := − Π kj =1 Res ar (∆ T j ( λ j . ) ) With these no-tations, the formula given in (i) takes the form F ( − λ ∆ T ) = ( − k (2 iπ ) k Z ˜ ∂U + ǫ F ( ζ ) R ( − λ ∆ T , ζ ) dζ. Clearly, F ( − λ ∆ T ) ∈ M ( B ) ⊂ QM r ( A ) . Let
U, U ′ ∈ W T,λ , let ( α, β ) and ( α ′ , β ′ ) be the elements of M a,b associated to U and U ′ and let ǫ ∈ S ∗ α,β and ǫ ′ ∈ S ∗ α ′ ,β ′ such that U + ǫ ∈ W T,λ and U ′ + ǫ ′ ∈ W T,λ . Set V = U + ǫ, V ′ = U ′ + ǫ ′ , V ′′ = V ∩ V ′ . Then the function G : ζ → F ( ζ ) R ( − λ ∆ T , ζ ) is holomorphic on a neighborhood of V \ V ′′ , and it follows from(43) that there exists M > such that | G ( ζ ) | ≤ M for ζ ∈ V \ V ′′ . The open sets V = Π j ≤ k V j and V ′′ = Π j ≤ k V ′′ j have the form ( z + S ∗ α,β ) \ K ) and ( z ′′ + S ∗ α ′′ ,β ′′ ) \ K ′′ ) where K and K ′′ are compact subsets of C k , and where α ′′ = inf( α, α ′ ) and β ′′ = sup ( β, β ′ ) . Choose ǫ ′′ ∈ S ∗ α ′′ ,β ′′ , and denote by V L,k the intersection of V k \ V ′′ k wtih the strip having for boundaries the lines D L = Le i ( − π − α k ) + R ǫ ′′ and D L = Le i ( π + β k ) + R ǫ ” . Set W n,j ( ζ j ) = n n +1+( ζ j − z j ) e i αj + βj ! , and set W n ( ζ ) = W n, ( ζ ) . . . W n,k ( ζ k ) . Then | W n ( ζ ) | < and lim n → + ∞ W n ( ζ ) = 1 for ζ ∈ V .
40t follows from Cauchy’s theorem that we have, when L is sufficiently large Z Π j ≤ k − ∂V j "Z ∂V L,k W n ( ζ ) G ( ζ ) dζ k dζ . . . dζ k − . We have, for s = 1 , , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z (Π j ≤ k − ∂V j ) × ( ∂V k ∩ D sL ) W n ( ζ ) G ( ζ ) dζ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M " Π j ≤ k − Z ∂V j | W n,j ( ζ j ) || dζ j | ∂V k ∩ D sL | W n,k || dζ k | , and so lim L → + ∞ R (Π j ≤ k − ∂V j ) × ( ∂V k ∩ D sL ) W n ( ζ ) G ( ζ ) dζ = 0 . We obtain Z ˜ ∂V W n ( ζ ) G ( ζ ) dζ = Z Π j ≤ k − ∂V j × ∂V ′′ k W n ( ζ ) G ( ζ ) dζ. It follows then from the Lebesgue dominated convergence theorem that wehave Z ˜ ∂V G ( ζ ) dζ = Z Π j ≤ k − ∂V j × ∂V ′′ k G ( ζ ) dζ. Using the same argument and a finite induction, we obtain Z ˜ ∂V G ( ζ ) dζ = Z ˜ ∂V ′′ G ( ζ ) dζ. Similarly R ˜ ∂V ′ G ( ζ ) dζ = R ˜ ∂V ′′ G ( ζ ) dζ, which shows that the definition of F ( − λ ∆ T , . . . , − λ k ∆ T k ) does not depend on the choice of U and ǫ. Now let F ∈ ∪ U ∈W T,λ H (1) ( U ) , let G ∈ ∪ U ∈W T,λ H (1) ( U ) . There exists U ∈W T,λ such that F | U ∈ H (1) ( U ) and G | U ∈ H (1) ( U ) . Choose ǫ ∈ S ∗ α,β , where ( α, β ) is the element of M a,b associated to U, such that U + ǫ ∈ W T,λ , and set V = U + ǫ , V ′ = U + ǫ. For M ⊂ { , . . . , k } , denote by | M | the cardinal of M. Then |{ , . . . , k } \ M | = 2 k − | M | . Since ( λ j ∆ T j + ζ j I ) − ( λ j ∆ T j + σ j I ) − = σ j − ζ j (cid:0) ( λ j ∆ T j + ζ j I ) − − ( λ j ∆ T j + σ j I ) − (cid:1) , we have F ( − λ ∆ T ) G ( − λ ∆ T ) = 1(2 iπ ) k Z ˜ ∂V × ˜ ∂V ′ F ( ζ ) G ( σ ) R ( − λ ∆ T , ζ ) R ( − λ ∆ T , σ ) dζdσ = 1(2 iπ ) k X M ⊂{ ,...,k } L M , where L M := ( − | M | Z ˜ ∂V × ˜ ∂V ′ σ − ζ . . . σ k − ζ k F ( ζ ) G ( σ )Π j ∈ M ( λ j ∆ T j + ζ j I ) − Π j ′ / ∈ M ( λ j ′ ∆ T j ′ + σ j ′ I ) − dζdσ. M = ∅ , and set W n,M (( σ j ) j ∈ M ) = Π j ∈ M n n +1+( σ j − z j ) e i αj + βj ! , where z ∈ C k is choosen so that z + S ∗ α,β ⊃ U. It follows from corollary 12.4 that G is bounded on V , and so the function ( σ j ) j ∈ M → W n,M (( σ j ) j ∈ M )Π j ∈ M σ j − ζ j G ( σ ) belongs to H (1) (Π j ∈ M V j + ( ǫ j ) j ∈ M ) for every ( σ j ) j / ∈ M and every ζ ∈ ˜ ∂V. Since the open set Π j ∈ M V j is admissible with respect to the family { ( α j , β j ) } j ∈ M , it follows from theorem 12.5 that we have, for every ( σ j ) j / ∈ M ∂V ′ j and every ζ ∈ ˜ ∂V Z Π j ∈ M ∂V j W n,M (( σ j ) j ∈ M )Π j ∈ M σ j − ζ j G ( σ )Π j ∈ M dσ j = 0 . Set P ( ζ, ( σ j ′ ) j ′ / ∈ M ) := R Π j ∈ M ∂V j Π j ∈ M σ j − ζ j G ( σ )Π j ∈ M dσ j . It follows then from the Lebesgue dominated convergence theorem that wehave, for ( σ j ) j / ∈ M ∈ Π j ′ / ∈ M ∂V ′ j and ζ ∈ ˜ ∂VP ( ζ, ( σ j ′ ) j ′ / ∈ M ) = 0 , and so ( − | M | L M = Z ˜ ∂V × (Π j ′ / ∈ M ∂V ′ j ′ ) Π j ′ / ∈ M σ j ′ − ζ j ′ F ( ζ ) P ( ζ, ( σ j ′ ) j ′ / ∈ M )Π j ∈ M ( λ j ∆ T j + ζ j I ) − Π j ′ / ∈ M ( λ j ′ ∆ T j ′ + σ j ′ I ) − dζ Π j ′ / ∈ M dσ j ′ = 0 . We obtain F ( − λ ∆ T ) G ( − λ ∆ T ) = 1(2 iπ ) k L ∅ = 1(2 iπ ) k Z ˜ ∂V ′ (cid:20)Z ˜ ∂V F ( ζ )( σ − ζ ) . . . ( σ k − ζ k ) dζ (cid:21) G ( σ )( λ ∆ + σ I ) − ( λ k ∆ k + σ k I ) − dσ = 1(2 iπ ) k Z ˜ ∂V ′ F ( σ ) G ( σ )( λ ∆ + σ I ) − ( λ k ∆ k + σ k I ) − dσ = ( F G )( − λ ∆ T ) , and so the map F → F ( − λ ∆ T ) is an algebra homomorphism from ∪ U ∈W T,λ H (1) ( U ) into M ( B ) . Let E be a bounded subset of ∪ U ∈W T,λ H (1) ( U ) , let U ∈ W T,λ such that E is abounded subset of H (1) ( U ) , let ( α, β ) be the element of M a,b associated to U, andlet ǫ ∈ S ∗ α,β be such that U + ǫ ∈ W T,λ . Set K = sup ζ ∈ ˜ ∂U + ǫ k R ( − λ ∆ T , ζ ) k M ( B ) . We have sup F ∈E k F ( − λ ∆ T ) k M ( B ) ≤ K (2 π ) k sup F ∈E k F k H (1) ( U ) < + ∞ , which shows that the map F → F ( − λ ∆ T ) is a bounded homomorphismfrom ∪ U ∈W T,λ H (1) ( U ) into M ( B ) ⊂ QM r ( A ) . U ∈ W T,λ , and let ( α, β ) ∈ M a,b and z ∈ C k be such that U ⊂ z + S ∗ α,β and ( z + S ∗ α,β ) \ U is bounded. For j ≤ k, set s j = 1 + sup lim t → + ∞ log (cid:16)(cid:13)(cid:13)(cid:13) T (cid:16) tλ j e i αj + βj (cid:17)(cid:13)(cid:13)(cid:13)(cid:17) t , − Re ( z j e i αj + βj . Set ˜ T j ( t ) = T ( tλ j e αj + βj ) for t > , with the convention ˜ T j (0) = I, and set, for f ∈ ∩ ζ ∈ C k e − ζ U α,β ,< f, φ > = Z ( R + ) k f ( t e i α β , . . . , t k e i αk + βk ) e − s t − ... − s k t k dt . . . dt k . Then z ∈ Dom ( FB ( φ )) , and we have, for ζ ∈ Dom ( FB ( φ )) , FB ( φ )( ζ ) = Z ( R + ) k e − t ζ e i α β ... − t k ζ k e i αk + βk e − s t + ... − s k t k dt . . . dt k = 1( ζ e i α β + s ) . . . ( ζ k e i αk + βk + s k ) ,< T ( λ ) , φ > = Z ( R + ) k T ( t λ e i α β ) . . . T k ( t λ k e i αk + βk ) e − s t − ... − s k t k dt . . . dt k = (cid:20)Z + ∞ ˜ T ( t ) e − s t dt (cid:21) . . . (cid:20)Z + ∞ ˜ T k ( t ) e − s k t k dt k (cid:21) , where the Bochner integrals are computed with respect to the strong oper-ator topology on M ( B ) . It follows from the observations in section 5 that (cid:16)R + ∞ ˜ T j ( t ) e − s j t dt j (cid:17) ( B ) is dense in B for ≤ j ≤ k, and so < T ( λ ) , φ > ( B ) is dense in B . Now set φ = φ ∗ φ. It follows from theorem 8.6 that we have < T ( λ ) , φ > = < T λ , φ > , and so < T ( λ ) , φ > ( B ) is dense in B . Set F = FB ( φ ) = FB ( φ ) . Then F ∈ H (1) ( U − ǫ ) ∩ H ∞ ( U − ǫ ) for some ǫ ∈ S ∗ α,β , and we have, using assertion (vi) of theorem 8.6 F ( − λ ∆ T ) = ( − k (2 iπ ) k Z z +˜ ∂S ∗ α,β F ( ζ )( λ ∆ T + σ ) − . . . ( λ k ∆ T k + σ k ) − dσ . . . dσ k = < T ( λ ) , φ >, F ( − λ ∆ T )( B ) is dense in B . Now consider again U ∈ W T,λ , and let F ∈ H ∞ ( U ) . Let G ∈ H (1) ( U ) besuch that G ( − λ ∆ T )( B ) is dense in B , and let u ∈ Ω( B ) . Then G ( − λ ∆ T ) u ∈ Ω( B ) , F G ∈ H (1) ( U ) , and so there exists a unique quasimultiplier R F ∈QM r ( B ) = QM r ( A ) such that R F G ( − λ ∆ T ) u = ( F G )( − λ ∆ T ) u, and R F = F ( − λ ∆ T ) if F ∈ H (1) ( U ) . Let U ′ ∈ W T,λ , and let G ∈ H (1) ( U ′ ) . We have R F G ( − λ ∆ T ) G ( − λ ∆ T ) = R F G ( − λ ∆ T ) G ( − λ ∆ T ) = ( F G )( − λ ∆ T ) G ( − λ ∆ T )= ( F G G )( − λ ∆ T ) = ( F G )( − λ ∆ T ) G ( − λ ∆ T ) , and so R F G ( − λ ∆ T ) = ( F G )( − λ ∆ T ) , which shows that the definition of R F does not depend on the choice of U. The map F → R F is clearly linear. Now let F ∈ ∪ U ∈W T,λ H ∞ ( U ) , let F ∈ ∪ U ∈W T,λ H ∞ ( U ) , and let G ∈ ∪ U ∈W T,λ H (1) ( U ) such that G ( − λ ∆ T )( B ) is dense in B . We have R F F G ( − λ ∆ T ) = ( F F G )( − λ ∆ T ) = ( F G )( − λ ∆ T )( F G )( − λ ∆ T )= R F R F G ( − λ ∆ T ) , and so R F F = R F R F since G ( − λ ∆ T ) B is dense in B . Now let E be a bounded family of elements of ∪ U ∈W T,λ H ∞ ( U ) . There exists U ∈ W T,λ and
M > such that F ∈ H ( ∞ ) ( U ) and k F k H ∞ ( U ) ≤ M for every F ∈ E . Let G ∈ H (1) ( U ) such that G ( − λ ∆ T )( B ) is dense in B . Then thefamily { F G } F ∈E is bounded in H (1) ( U ) , and it follows from (i) that there exists u ∈ Ω( B ) such that sup F ∈E k ( F G )( − λ ∆ T ) u k B < + ∞ . We obtain sup F ∈E k R F G ( − λ ∆ T ) u k B = sup F ∈E k ( F G )( − λ ∆ T ) u k B < + ∞ , and so the family { R F } F ∈E is pseudobounded in QM ( B ) = QM ( A ) since G ( − λ ∆ T ) u ∈ Ω( B ) . Since the family { λ − n F n } n ≥ is bounded in H ∞ ( U ) for F ∈ H ∞ ( U ) , λ > (1+ k F k H ∞ ( U ) ) − , this shows that R F ∈ QM r ( B ) = QM r ( A ) for F ∈ ∪ U ∈F H ∞ ( U ) , and that the map F → R F is a bounded algebra homo-morphism from ∪ U ∈F H ∞ ( U ) into QM r ( B ) = QM r ( A ) , which concludes theproof of (ii).(iii) Let ( α, β ) ∈ N λ , let φ ∈ F α,β , assume that N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ , and let z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) . Then z + S ∗ α,β is admissible withrespect to ( T, λ, α, β ) . As in the proof of (ii) we can construct φ ∈ F α,β havingthe following properties • z ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) , • G := FB ( φ ) ∈ H (1) ( z + S ∗ α,β ) ∩ H ∞ ( z + S ∗ α,β ) , • < T ( λ ) , φ > = G ( − λ ∆ T ) , and G ( − λ ∆ T )( B ) is dense in B . ǫ ∈ S ∗ α,β be such that z + ǫ + S ∗ α,β is admissible with respect to ( T, λ, α, β ) . It follows from assertions (v) and (vi) of theorem 8.6 and from (i) and (ii) thatwe have < T ( λ ) , φ > FB ( φ )( − λ ∆ T ) = < T λ ) , φ >< T ( λ ) , φ > = < T ( λ ) , φ ∗ φ > = Z z + ǫ + S ∗ α,β FB ( φ )( σ ) FB ( φ )( σ )( λ ∆ T + σ I ) − . . . ( λ k ∆ T k + σ k I ) − dσ . . . dσ k = ( FB ( φ ) FB ( φ ))( − λ ∆ T ) = FB ( φ )( − λ ∆ T ) FB ( φ )( − λ ∆ T ) , and so < T ( λ ) , φ > = FB ( φ )( − λ ∆ T ) since FB ( φ )( − λ ∆ T )( B ) is dense in B . Now let ν ∈ C such that νλ j ∈ ∪ ( γ j ,δ j ) ∈ M aj,bj S γ j ,δ j , and let ν j = ( ν j, , . . . , ν j,k ) be the k -tuple defined by the conditions ν j,s = 0 if s = j, ν j,j = ν. There exists ( γ j , δ j ) ∈ N λ j such that ν ∈ S γ j ,δ j , and there exists ( α, β ) ∈ N λ such that α j = γ j and β j = δ j . Set F ( ζ ) = e − νζ j for ζ ∈ C k , and set < f, φ > = f ( ν j ) for f ∈ ∩ z ∈ C k e − z U α,β . Then
Dom ( FB ( φ )) = C k , and we have, for ζ ∈ C k , FB ( φ )( ζ ) = < e − ζ , φ > = e − ν j ζ = e − νζ j , and so F = FB ( φ ) . Let z ∈ N ( T, λ, α, β ) = N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) . Let δ ν j be the Dirac measure at ν j . Since e − z δ ν j is a representing measure for φe − z we have F ( − λ ∆ T ) = < T ( λ ) , φ > = T j ( νλ j ) , which concludes the proof of (iii).(iv) Let ( α, β ) ∈ N λ , let φ ∈ F α,β , and assume that N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) = ∅ . Set e − ǫ T = ( e − ǫ T , . . . , e − ǫ k T k ) . Then N ( T, λ, α, β ) ⊂ N ( e − ǫ T, λ, α, β ) for ǫ ∈ S ∗ α,β , and it follows from theorem 8.6 (ii) and from (iii) that we have, for u ∈ B , < T ( λ ) , φ > u = lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β < e − ǫ T λ , φ > u = lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β FB ( φ )( e − ǫ T ( λ ) ) u = lim ǫ → (0 ,..., ǫ ∈ S ∗ α,β FB ( φ )( − λ T + ǫ I, . . . , − λ k T k + ǫ k I ) u, which concludes the proof of (iv).(v) Let U ∈ W t,λ , let F ∈ H ∞ ( U ) be strongly outer, and let ( F n ) n ≥ be asequence of invertible elements of H ∞ ( U ) satisfying the conditions of definition12.6 with respect to F. It follows from (ii) that there exists G ∈ H (1) ( U ) ∩ H ∞ ( U ) such that G ( − λ ∆ T )( B ) is dense in B . Let ( α, β ) ∈ M a,b and z ∈ N ( T, λ, α, β ) such that U ⊂ z + S ∗ α,β and such that ( z + S ∗ α,β ) \ U is bounded. There exists ǫ ∈ C k such that U + ǫ ⊂ U is admissible with respect to ( T, λ, α, β ) and wehave F ( − λ ∆ T ) F − n ( − λ ∆ T ) G ( − λ ∆ T ) iπ ) k Z ǫ +˜ ∂U F ( σ ) F − n ( σ ) G ( σ )( λ ∆ T + σ I ) − . . . ( λ k ∆ T k + σ k I ) − dσ . . . dσ k , and it follows from the Lebesgue dominated convergence theorem that lim n → + ∞ k F ( − λ ∆ T ) F − n ( − λ ∆ T ) G ( − λ ∆ T ) − G ( − λ ∆ T ) k M ( B ) = 0 . Let u ∈ Ω( B ) . Then G ( − λ ∆ T ) u ∈ Dom ( F ( − λ ∆ T )) ∩ Ω( B ) . Set u n = F − n ( − λ ∆ T ) G ( − λ ∆ T ) u ∈ B . We have G ( − λ ∆ T ) u = lim n → + ∞ F ( − λ ∆ T ) G ( − λ ∆ T ) uu n . Since G ( − λ ∆) u ∈ Ω( B ) , this shows that F ( − λ ∆ T ) G ( − λ ∆ T ) u ∈ Ω( B ) , whichproves (v).(vi) Let U ∈ W T,λ , let F ∈ S ( U ) , let G ∈ H ∞ ( U ) be a strongly outerfunction such that F G ∈ H ∞ ( U ) , and let u ∈ Dom ( G ( − λ ∆ T )) such that G ( − λ ∆ T ) u ∈ Ω( B ) . Let v ∈ Ω( B ) ∩ Dom ( F G ( − λ ∆ T )) . There exists a unique R F ∈ QM ( B ) = QM ( A ) satisfying the equation ( F G )( − λ ∆ T ) uv = R F G ( − λ ∆ T ) uv, and we have ( F G )( − λ ∆ T ) = R F G ( − λ ∆ T ) , so that R F = F ( − λ ∆ T ) if F ∈ H ∞ ( U ) . Let G ∈ ∪ V ∈W T,λ H ∞ ( V ) such that F G ∈ H ∞ ( W ) for some W ∈ W T,λ , andlet w ∈ Ω( B ) ∩ Dom ( G ( − λ ∆ T )) . We have (( F G )( − λ ∆ T ) vw ) G ( − λ ∆ T ) u = ( F G )( − λ ∆ T ) G ( − λ ∆ T ) uvw = R F G ( − λ ∆ T ) G ( − λ ∆ T ) uvw = ( R F G ( − λ ∆ T ) vw ) G ( − λ ∆ T ) u. Since vw ( G ( − λ ∆ T ) u ) ∈ Ω( B ) , this shows that ( F G )( − λ ∆ T ) = R F G ( − λ ∆ T ) . So if we set F ( − λ ∆ T ) = R F , we obtain F ( − λ ∆ T ) G ( − λ ∆ T ) = ( F G )( − λ ∆ T ) for every F ∈ ∪ U ∈W T,λ S ( U ) and for every G ∈ ∪ U ∈W T,λ H ∞ ( U ) such that F G ∈ ∪ U ∈W T,λ H ∞ ( U ) . The map F → F ( − λ ∆ T ) is clearly linear. Now let F ∈∪ U ∈W T,λ S ( U ) , F ∈ ∪ U ∈W T,λ S ( U ) , and let G ∈ ∪ U ∈W T,λ H ∞ ( U ) and G ∈∪ U ∈W T,λ H ∞ ( U ) be strongly outer functions such that F G ∈ ∪ U ∈W T,λ H ∞ ( U ) and F G ∈ ∪ U ∈W T,λ H ∞ ( U ) . We have ( F F )( − λ ∆ T ) G ( − λ ∆ T ) G ( − λ ∆ T ) = ( F F G G )( − λ ∆ T )= ( F G )( − λ ∆ T )( F G )( − λ ∆ T ) = F ( − λ ∆ T ) F ( − λ ∆ T ) G ( − λ ∆ T ) G ( − λ ∆ T ) , and so ( F F )( − λ ∆ T ) = F ( − λ ∆ T ) F ( − λ ∆ T ) since Dom ( G ( − λ ∆ T )) ∩ Ω( B ) = ∅ and Dom ( G ( − λ ∆ T )) ∩ Ω( B ) = ∅ , and the map F → F ( − λ ∆ T ) is analgebra homomorphism from ∪ U ∈W T,λ S ( U ) into QM ( B ) = QM ( A ) . Now let E be a bounded family of elements of ∪ U ∈W T,λ S ( U ) . There ex-ists U ∈ W T,λ and a strongly outer function G ∈ H ∞ ( U ) such that F G ∈ ∞ ( U ) for every F ∈ E and such that sup F ∈E k F G k H ∞ ( U ) < + ∞ . So the fam-ily { ( F G )( − λ ∆ T ) } F ∈E is a pseudobounded family of elements of QM r ( B ) = QM r ( A ) , and there exists u ∈ Ω( B ) ∩ ( ∩ F ∈E Dom (( F G )( − λ ∆ T ))) such that sup F ∈E k ( F G )( − λ ∆ T ) u k B < + ∞ . Let v ∈ Dom ( G ( − λ ∆ T )) ∩ Ω( B ) , and set w = G ( − λ ∆ T ) uv. Then w ∈ Ω( B ) ∩ ( ∩ F ∈E Dom ( F ( − λ ∆ T )) and sup F ∈E k F ( − λ ∆ T ) w k B = sup F ∈E k ( F ( − λ ∆ T ) G ( − λ ∆ T ) uv k B ≤ sup F ∈E k ( F G )( − λ ∆ T ) u k B k v k B < + ∞ , and so the family { F ( − λ ∆ T } F ∈E is pseudobounded in QM ( B ) = QM ( A ) , andthe map F → F ( − λ ∆ T ) is a bounded algebra homomorphism from ∪ U ∈W T,λ S ( U ) into QMB ) = QM ( A ) , which concludes the proof of (vi).Now assume that A is not radical, let χ ∈ b A , and let ˜ χ be the uniquecharacter on QM ( A ) such that ˜ χ ( u ) = χ ( u ) for every u ∈ A . Let F ∈ H (1) ( U ) , where U ∈ W T,λ , let ( α, β ) be the element of M a,b asso-ciated to U, and let ǫ ∈ S ∗ α,β be such that U + ǫ is admissible with respect to ( T, λ, α, β ) . Since Bochner integrals commute with linear functionals, we have ˜ χ ( F ( − λ ∆ T , . . . , − λ k ∆ T k ))= 1(2 iπ ) k Z ˜ ∂U + ǫ F ( ζ , . . . , ζ k )( λ ˜ χ (∆ T )+ ζ I ) − . . . ( λ ˜ χ (∆ T k )+ ζ k I ) − dζ . . . dζ k . Since U + ǫ is admissible with respect to ( T, λ, α, β ) , ( − λ ˜ χ (∆ T ) , . . . , − λ k ˜ χ (∆ T k )) ∈ U + ǫ, and it follows from theorem 12.5 that we have ˜ χ ( F ( − λ ∆ T , . . . , − λ k ∆ T k )) = F ( − λ ˜ χ (∆ T ) , . . . , − λ k ˜ χ (∆ T k )) . Now let F ∈ H ∞ ( U ) , where U ∈ W T,λ , and let G ∈ H (1) ( U ) such that G ( − λ ∆ T )( B ) is dense in B . Then ˜ χ ( G ( − λ ∆ T )) = 0 , and we have ˜ χ ( F ( − λ ∆ T )) = ˜ χ (( F G )( − λ ∆ T ))˜ χ ( G ( − λ ∆ T )) = ( F G )( − λ ˜ χ (∆ T )) G ( − λ ˜ χ (∆ T )) = F ( − λ ˜ χ (∆ T )) . Finally let F ∈ S ( U ) , where U ∈ W T,λ , and let G ∈ H ∞ ( U ) be a stronglyouter function such that F G ∈ H ∞ ( U ) . It follows from (v) that G ( − λ ∆ T ) u ∈ Ω( B ) for some u ∈ B , and so ˜ χ ( G ( − λ ∆ T )) = 0 . The same argument as aboveshows then that ˜ χ ( F ( − λ ∆ T )) = F ( − λ ˜ χ (∆ T )) , which concludes the proof of(vi).(vii) Set F ( ζ , . . . , ζ k ) = − ζ j , choose ν > ν > lim t → + ∞ log k T j ( tλ j ) k t , andset again v ν ( t ) = te − ν t . It follows from proposition 12.8(ii) that F ∈ S ( U ) forevery U ∈ W T,λ , and it follows from proposition 5.5(i) that we have λ j ∆ T j Z [0 , ∞ ) n v ν ( t ) T j ( tλ j ) dt = − Z + ∞ v ′ ν ( t ) T j ( tλ j ) dt, M ( B ) . Now choose ( α, β ) ∈ N λ , and set, for f ∈ ∩ z ∈ C k e − z U α,β ,< f, φ > = Z [0 , + ∞ ) k f (0 , . . . , , t j , , . . . , v ν ( t j ) dt j ,< f, φ > = Z [0 , + ∞ ) k f (0 , . . . , , t j , , . . . , v ′ ν ( t j ) dt j . Then φ ∈ F α,β , φ ∈ F α,β , − ν λ j + S ∗ α,β ∈ N ( T, λ, α, β ) ∩ Dom ( FB ( φ )) ∩ Dom ( FB ( φ )) , and it follows from (iii) that we have Z [0 , ∞ ) n v ν ( t ) T j ( tλ j ) dt j = < T ( λ ) , φ > = FB ( φ )( − λ ∆ T ) , Z [0 , ∞ ) n v ′ ν ( t ) T j ( tλ j ) dt j = < T ( λ ) , φ > = FB ( φ )( − λ ∆ T ) . But FB ( φ )( ζ ) = ν + ζ j , FB ( φ )( ζ ) = ζ j ν + ζ j = − F ( ζ ) FB ( φ )( ζ ) , whichgives λ j ∆ T j Z [0 , ∞ ) n v ν ( t ) T j ( tλ j ) dt = F ( − λ ∆ T ) Z [0 , ∞ ) n v ν ( t ) T j ( tλ j ) dt, and so F ( − λ ∆ T ) = λ j ∆ T j , since ( R [0 , ∞ ) n v ν ( t ) T j ( tλ j ) dt )( B ) is dense in B , as observed in section 5. (cid:3)
10 Appendix 1: Fourier-Borel and Cauchy trans-forms
In this section we present some certainly well-known results about Fourier-Boreland Cauchy transforms of linear functionals on some spaces of holomorphicfunctions on sectors.For α < β ≤ α + π denote as usual by S α,β the closure of the open sector S α,β , and set by convention S α,α := { te iα } t ≥ . We set S ∗ α,β = S − π/ − α,π/ − β , S ∗ α,β = S − π/ − α,π/ − β . (15)Notice that S ∗ α,α + π = ∅ , while S ∗ α,α + π = S − π/ − α, − π/ − α = {− tie − iα } t ≥ Now asssume that α ≤ β < α + π. Let λ = | λ | e iω ∈ S ∗ α,β and let ζ = | ζ | e iθ ∈ S α,β , with − π − α ≤ ω ≤ π − β, α ≤ θ ≤ β. We have − π ≤ ω + θ ≤ π , | e − λζ | = e −| λ || ζ | cos ( ω + θ ) , and we obtain | e − λζ | < λ ∈ S ∗ α,β , ζ ∈ S α,β \ { } ) . (16) | e − λζ | ≤ λ ∈ S ∗ α,β , ζ ∈ S α,β ) . (17)48 efinition 10.1 Let α = ( α , . . . , α k ) , β = ( β , . . . β k ) ∈ R k such that α j ≤ β j < α j + π for ≤ j ≤ k. Set S α,β := Π kj =1 S α j ,β j , S ∗ α,β := Π kj =1 S ∗ α j ,β j ,S ∗ α,β := Π kj =1 S ∗ α j ,β j . If, further, α j < β j for ≤ j ≤ k, set S α,β := Π kj =1 S α j ,β j . Let X be a Banach space. We denote by U α,β ( X ) the set of all continuous X -valued functions f on S α,β satisfying lim | z |→ + ∞ z ∈ Sα,β k f ( z ) k X = 0 such that themap ζ → f ( ζ , ζ , . . . , ζ j − , ζ, ζ j +1 , . . . , ζ k ) is holomorphic on S α j ,β j for every ( ζ , . . . , ζ j − , ζ j +1 , . . . , ζ k ) ∈ Π ≤ s ≤ ks = j S α s ,β s when α j < β j . Similarly we denote by V α,β ( X ) the set of all continuous bounded X -valuedfunctions f on S α,β such that the map ζ → f ( ζ , ζ , . . . , ζ j − , ζ, ζ j +1 , . . . , ζ k ) is holomorphic on S α j ,β j for every ( ζ , . . . , ζ j − , ζ j +1 , . . . , ζ k ) ∈ Π ≤ s ≤ ks = j S α s ,β s when α j < β j . The spaces U α,β ( X ) and V α,β ( X ) are equipped with the norm k f k ∞ = sup z ∈ S α,β k f ( z ) k X , and we will write U α,β := U α,β ( C ) , V α,β := V α,β ( C ) . A representing measure for φ ∈ U ′ α,β is a measure of bounded variation ν on S α,β satisfying < f, φ > = Z S α,β f ( ζ ) dν ( ζ ) ( f ∈ U α,β ) . (18) . Set I := { j ≤ k | α j = β j } , J := {{ j ≤ k | α j < β j } . Since separate holo-morphy with respect to each of the variables z j , j ∈ J implies holomorphy withrespect to z J = ( z j ) j ∈ J , the map z J → f ( z I , z J ) is holomorphic on Π j ∈ J S α j ,β j for every z I ∈ Π j ∈ I S α j ,α j . For z = ( z , . . . , z k ) , ζ = ( ζ , . . . , ζ k ) ∈ C k , set again e z ( ζ ) = e z ζ ... + z k ζ k . Also set, if X is a separable Banach space, and if α = ( α , . . . , α k ) and β =( β , . . . , β k ) satisfy the conditions above U ∗ α,β ( X ) = U ( − π/ − α ,..., − π/ − α k ) , ( π/ − β ,...,π/ − β k ) ( X ) , (19) V ∗ α,β ( X ) = V ( − π/ − α ,..., − π/ − α k ) , ( π/ − β ,...,π/ − β k ) ( X ) , (20)with the conventions U ∗ α,β = U ∗ α,β ( C ) , V ∗ α,β = V ∗ α,β ( C ) . Proposition 10.2
Let φ ∈ U ′ α,β , and let X be a separable Banach space. Set,for f ∈ V α,β ( X ) , < f, φ > = Z S α,β f ( ζ ) dν ( ζ ) , where ν is a representing measure for φ. Then this definition does not dependon the choice of ν, and we have < f, φ > = lim ǫ → ǫ ∈ S ∗ α,β < e − ǫf , φ > . (21)49roof: It follows from (16) and (17) that e − ǫ f ∈ U α,β ( X ) for f ∈ V α,β ( X ) , ǫ ∈ S ∗ α,β . If f ∈ U α,β ( X ) , then we have, for l ∈ U α,β ( X ) ′ ,< Z S α,β f ( ζ ) dν ( ζ ) , l > = Z S α,β < f ( ζ ) , l > dν ( ζ ) = << f ( ζ ) , l >, φ >, which shows that the definition of < f, φ > does not depend on the choiceof ν. Now if f ∈ V α,β ( X ) , it follows from the Lebesgue dominated convergencetheorem that we have Z S α,β f ( ζ ) dν ( ζ ) = lim ǫ → ǫ ∈ S ∗ α,β Z S α,β e − ǫ ( ζ ) f ( ζ ) dν ( ζ ) = lim ǫ → ǫ ∈ S ∗ α,β < e − ǫ f, φ >, and we see again that the definition of < f, φ > does not depend on thechoice of the measure ν. (cid:3) We now introduce the classical notions of Cauchy transforms and Fourier-Borel transforms.
Definition 10.3
Let φ ∈ U ′ α,β , and let f ∈ V α,β ( X ) .(i) The Fourier-Borel transform of φ is defined on S ∗ α,β by the formula FB ( φ )( λ ) = < e − λ , φ > ( λ ∈ S ∗ α,β ) . (ii) The Cauchy transform of φ is defined on Π ≤ j ≤ k ( C \ S α j ,β j ) by theformula C ( φ )( λ ) = 1(2 iπ ) k < ζ − λ ) , φ ζ > := 1(2 iπ ) k < ζ − λ . . . ζ k − λ k , φ ζ ,...,ζ k > ( λ = ( λ , . . . , λ k ) ∈ Π ≤ j ≤ k ( C \ S α j ,β j )) . (iii) The Fourier-Borel transform of f is defined on Π ≤ j ≤ k ( C \ − S ∗ α j ,β j ) bythe formula FB ( f )( λ ) = Z e iω . ∞ e − λζ f ( ζ ) dζ := Z e iω . ∞ . . . Z e iωk . ∞ e − λ ζ − ... − λ k ζ k f ( ζ , . . . , ζ k ) dζ . . . dζ k ( λ = ( λ , . . . , λ k ) ∈ Π ≤ j ≤ k ( C \ − S ∗ α j ,β j )) , where α j ≤ ω j ≤ β j and where Re ( λ j e iω j ) > for ≤ j ≤ k.
50t follows from these definitions that C ( φ ) is holomorphic on Π ≤ j ≤ k ( C \ S α j ,β j ) for φ ∈ U ′ α,β , and that FB ( f ) is holomorphic on Π ≤ j ≤ k ( C \ − S ∗ α j ,β j ) for f ∈ V α,β ( X ) . Also using proposition 10.2 we see that FB ( φ ) ∈ V ∗ α,β := V − π − α, π − β for φ ∈ U ′ α,β . Proposition 10.4
Let φ ∈ U ′ α,β . For j ≤ k, set I η,j = ( π − η, π − β j ] for η ∈ ( β j , α j + π ] , I η,j = ( − π − α j , π − β j ) for η ∈ ( α j + π, β j + π ] , and set I η = ( − π − α j , π − η ) for η ∈ ( β j + π, α j + 2 π ) . Then I η,j ⊂ [ − π − α j , π − β j ] ,cos ( η + s ) < for s ∈ I η,j , and if λ = ( λ , . . . , λ k ) ∈ Π ≤ j ≤ k ( C \ S α j ,β j ) , wehave for ω = ( ω , . . . , ω k ) ∈ Π ≤ j ≤ k I arg ( λ j ) ,j , C ( φ )( λ ) = 1(2 iπ ) k Z e iω . ∞ e λσ FB ( φ )( σ ) dσ := 1(2 πi ) k Z e iω . ∞ . . . Z e iωk . ∞ e λ σ + ... + λ k σ k FB ( φ )( σ , . . . , σ k ) dσ . . . dσ k . (22)Proof: It follows from the definition of I η,j that I η,j ⊂ [ − π − α j , π − β j ] . Inthe second case we have obviously π < η + s < π for s ∈ I η j . In the first casewe have π < η + s < π + η − β j ≤ π + α j − β j < π for ω ∈ I η,j and in thethird case we have π > η + s > π + β j − π − α j > π for ω ∈ I η,j . We thus seethat cos ( η + s ) < for η ∈ ( β j , π + α j ) , ω ∈ I η,j . Now assume that λ ∈ Π ≤ j ≤ k ( C \ S α j ,β j ) , let η j ∈ ( β j , π + α j ) be a determi-nation of arg ( λ j ) , let ν be a representing measure for φ and let ω ∈ Π ≤ j ≤ k I η j . Then FB ( φ ) is bounded on S − π − α, π − β , and since cos ( η j + ω j ) < for j ≤ k, we have iπ ) k Z e iω . ∞ e λσ FB ( φ )( σ ) dσ = 1(2 iπ ) k Z e iω . ∞ e λσ "Z S α,β e − σζ dν ( ζ ) dσ = 1(2 iπ ) k Z S α,β "Z e iω . ∞ e σ ( λ − ζ ) dσ dν ( ζ ) = Z S α,β ζ − λ dν ( ζ ) = C ( φ )( λ ) . (cid:3) Now identify the space M ( S α,β ) of all measures of bounded variation on S α,β to the dual space of the space C ( S α,β ) of continuous functions on S α,β vanishingat infinity via the Riesz representation theorem.The convolution product of twoelements of M ( S α,β ) is defined by the usual formula Z S α,β f ( ζ ) d ( ν ∗ ν )( ζ ) := Z S α,β × S α,β f ( ζ + ζ ′ ) dν ( ζ ) dν ( ζ ′ ) ( f ∈ C ( S α,β )) . Proposition 10.5
Let X be a separable Banach space. i) For f ∈ V α,β ( X ) , λ ∈ S α,β , set f λ ( ζ ) = f ( ζ + λ ) . Then f λ ∈ V α,β ( X ) for f ∈ V α,β ( X ) , f λ ∈ U α,β ( X ) and the map λ → f λ belongs to U α,β ( U α,β ( X )) for f ∈ U α,β ( X ) . Moreover if we set, for φ ∈ U ′ α,β ,f φ ( λ ) = < f λ , φ >, then f φ ∈ V α,β ( X ) for f ∈ V α,β ( X ) , and f φ ∈ U α,β ( X ) for f ∈ U α,β ( X ) . (ii) For φ ∈ U ′ α,β , φ ∈ U ′ α,β , set < f, φ ∗ φ > = < f φ , φ > ( f ∈ U α,β ) . Then φ ∗ φ ∈ U ′ α,β , ν ∗ ν is a representing measure for φ ∗ φ if ν is arepresenting measure for φ and if ν is a representing measure for φ , and wehave < f, φ ∗ φ > = < f φ , φ > ( f ∈ V α,β ( X )) , FB ( φ ∗ φ ) = FB ( φ ) FB ( φ ) . Proof: These results follow from standard easy verifications which are leftto the reader. We will just prove the last formula. Let φ ∈ U ′ α,β , φ ∈ U ′ α,β . We have, for z = ( z , . . . , z k ) ∈ S ∗ α,β , λ = ( λ , . . . , λ k ) ∈ S α,β , ζ = ( ζ , . . . , ζ k ) ∈ S α,β , ( e − z ) λ ( ζ ) = e − z ( λ + ζ ) ... − z k ( λ k + ζ k ) = e − z ( λ ) e − z ( ζ ) , and so ( e − z ) λ = e − z ( λ ) e − z , ( e − z ) φ ( λ ) = < ( e − z ) λ , φ > = e − z ( λ ) FB ( φ )( z ) , ( e − z ) φ = FB ( φ )( z ) e − z , and FB ( φ ∗ φ )( z ) = < ( e − z ) φ , φ > = FB ( φ )( z ) < e − z , φ > = FB ( φ )( z ) FB ( φ )( z ) . (cid:3) For η ∈ S α,β , denote by δ η the Dirac measure at η. We identify δ η to thelinear functional f → f ( η ) on U α,β . With the above notations, we have, for f ∈ V α,β ( X ) , φ ∈ U ′ α,β ,f δ η = f η , < f, φ ∗ δ η > = < f η , φ > . If f ∈ U α,β ( X ) , we have lim ǫ → ǫ ∈ S ∗ α,β k e − ǫ f − f k ∞ = 0 = lim η → η ∈ Sα,β k f η − f k ∞ . We obtain, since k e − ǫ f k ∞ ≤ k f k ∞ for f ∈ U α,β , ǫ ∈ S ∗ α,β , lim ǫ → ,ǫ ∈ S ∗ α,βη → ,η ∈ Sα,β k e − ǫ f η − f k ∞ = 0 ( f ∈ U α,β ( X ) . (23)Now let f ∈ V α,β ( X ) , let φ ∈ U ′ α,β , and let ν be a representative measure for φ. Since < e ǫ f η > = R S α,β e − ǫζ f ( ζ + η ) dν ( ζ ) , and since < e − ǫ f, φ ∗ δ η > = e − ǫη < − ǫ f η , φ >, it follows from the Lebesgue dominated convergence theorem thatwe have < f, φ > = lim ǫ → ,ǫ ∈ S ∗ α,βη → ,η ∈ Sα,β < e − ǫ f η , φ > = lim ǫ → ,ǫ ∈ S ∗ α,βη → ,η ∈ Sα,β < e − ǫ f, φ ∗ δ η > ( f ∈ V α,β ( X ) , φ ∈ U ′ α,β ) . (24)In the following we will denote by ˜ ∂S α,β = Π ≤ j ≤ k ∂S α j ,β j the distinguishedboundary of S α,β , where ∂S α j ,β j = ( e iα j . ∞ , ∪ [0 , e iβ j . ∞ ) is oriented from e iα j . ∞ towards e iβ j . ∞ . The following standard computations allow to compute in some cases
Assume that α j < β j < α j + π for ≤ j ≤ k, and let φ ∈ U ′ α,β . If f ∈ V α,β ( X ) , and if Z ˜ ∂S α,β k f ( σ ) k X | dσ | < + ∞ , then we have, for η ∈ S α,β ,< f η , φ > = < f, φ ∗ δ η > = Z ˜ ∂S α,β C ( φ )( σ − η ) f ( σ ) dσ. (25) In particular we have, for f ∈ V α,β ( X ) , ǫ ∈ S ∗ α,β , η ∈ S α,β ,e − ǫη < e − ǫ f η , φ > = < e − ǫ f, φ ∗ δ η > = Z ˜ ∂S α,β e − ǫσ C ( φ )( σ − η ) f ( σ ) dσ (26)Proof: Assume that f ∈ V α,β ( X ) satisfies the condition sup σ ∈ S α,β (1 + | σ | ) k k f ( σ ) k < + ∞ . Let ν ∈ M ( S α,β ) be a representing measure for φ. For
R > , j ≤ k, wedenote by Γ R,j the Jordan curve { Re iω } α j ≤ ω ≤ β j ∪ [ Re iβ j , ∪ [0 , Re iα j ] , orientedcounterclockwise.We have, for η ∈ S α,β , σ ∈ Π ≤ j ≤ k ∂S α j ,β j , |C ( φ )( σ − η ) | ≤ π ) k k φ k U ′ α,β Π ≤ j ≤ k dist ( ∂S α j ,β j − η j , ∂S α j ,β j ) − . It follows then from Fubini’s theorem and Cauchy’s formula that we have Z ˜ ∂S α,β C ( φ )( σ − η ) f ( σ ) dσ = Z S α,β " iπ ) k Z ˜ ∂S α,β f ( σ ) ζ − σ + η dσ dν ( ζ )= Z S α,β lim R → + ∞ iπ ) k "Z Γ R, . . . Z Γ R,k f ( σ )( σ − ζ − η ) . . . ( σ k − ζ k − η k ) dσ dν ( ζ ) Z S α,β f ( ζ + η ) dν ( ζ ) = < f, φ ∗ δ η > . Formula (26) follows from this equality applied to e − ǫ f. Taking the limit as ǫ → , ǫ ∈ S ∗ α,β in formula (26), we deduce formula (25) from the Lebesguedominated convergence theorem. (cid:3) The following result is indeed standard, but we give a proof for the conve-nience of the reader.
Proposition 10.7
The linear span of the set E α,β := { f = e − σ : σ ∈ Π j ≤ k (0 , e − i αj + βj . ∞ ) } is dense in U α,β , and the Fourier-Borel transform is one-to-one on U ′ α,β . Proof: Set J = { j ∈ { , . . . , k } | α j = β j } , set J := { j ∈ { , . . . , k } | α j <β j } , denote by U the space of continuous functions on S = Π j ∈ J S α j ,β j vanish-ing at infinity, set S := Π j ∈ J S α j ,β j , and denote by U the space of continuousfunctions on S vanishing at infinity which satisfy the same analyticity condi-tion as in definition 10.1 with respect to S . Also set E := { f = e − σ : σ ∈ Π j ∈ J (0 , e − i αj + βj . ∞ ) } , and set E := { f = e − σ : σ ∈ Π j ∈ J (0 , e − i αj + βj . ∞ ) } . Assume that J = ∅ . Then the complex algebra span ( E ) is self-adjoint andseparates the point on U , and it follows from the Stone-Weierstrass theoremapplied to the one-point compactification of S that span ( E ) ⊕ C . is dense in U ⊕ C . , which implies that span ( E ) is dense in U since U is the kernel of acharacter on U ⊕ C . . Now assume that J = ∅ , set S ∗ = Π j ∈ J S − π − α j , π − β j , let φ ∈ U ′ , and definethe Cauchy transform and the Fourier-Borel transform of φ as in definition 10.3.Assume that < f, φ > = 0 for φ ∈ E . If j ∈ J , then g = 0 for every holomorphicfunction g on S ∗ α j ,β j which vanishes on (0 , e − i αj + βj . ∞ ) . An immediate finiteinduction shows then that FB ( φ ) = 0 since FB ( φ ) is holomorphic on S ∗ . Itfollows then from proposition 10.4 that C ( φ ) = 0 , and it follows from (23) and(26) that < f, φ > = 0 for every f ∈ U . Hence φ = 0 , which shows that span ( E ) is dense in U . This shows that span ( E α,β ) is dense in U α,β if J = ∅ or if J = ∅ . Now assume that J = ∅ and J = ∅ , and denote by E ⊂ U α,β the setof products f = gh, where g ∈ U and h ∈ U . The space U = C ( S ) is aclosed subsbace of codimension one of C ( S ∪ {∞} ) . Since the space C ( K ) has aSchauder basis for every compact space K , [3],[30], the space U has a Schauderbasis. Identifying the dual space of U to the space of measures of boundedvariation on S , this means that there exists a sequence ( g n ) n ≥ of elements of U and a sequence ( ν n ) n ≥ of measures of bounded variation on S such thatwe have g = + ∞ X n =1 (cid:18)Z S g ( η ) dν n ( η ) (cid:19) g n ( g ∈ U ) , where the series is convergent in ( U , k . k ∞ ) . Set P m ( g ) = m P n =1 (cid:16)R S g ( η ) dν n ( η ) (cid:17) g n for g ∈ U , m ≥ . Then P m : U → U is a bounded linear operator, and lim sup m → + ∞ k P m ( g ) k ≤ k g k < + ∞ for54very g ∈ U . It follows then from the Banach-Steinhaus theorem that thereexists
M > such that k P m k B ( U ) ≤ M for m ≥ , a standard property ofSchauder bases in Banach spaces.Now let φ ∈ U ′ α,β such that < f, φ > = 0 for f ∈ E, let ν be a repe-senting measure for φ, and let f ∈ U α,β . The function f ζ = η → f ( η, ζ ) belongs to U for ζ ∈ S , and a routine verification shows that the function h n : ζ → R S f ζ ( σ ) dν n ( σ ) = R S f ( ζ, η ) dν n ( η ) belongs to U for n ≥ . Sincethe evaluation map g → g ( η ) is continuous on U for η ∈ S , we obtain, for η ∈ S , ζ ∈ S , f ( η, ζ ) = lim m → + ∞ m X n =1 g n ( η ) h n ( ζ ) . We have, for m ≥ , η ∈ S , ζ ∈ S , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X n =1 g n ( η ) h n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k P m ( f ζ ) k ∞ ≤ M k f ζ k ∞ ≤ M k f k ∞ . It follows then from the Lebesgue dominated convergence theorem that Z S α,β f ( η, ζ ) dν ( η, ζ ) = lim m → + ∞ m X n =1 Z S α,β g n ( η ) h n ( ζ ) dν ( η, ζ ) = 0 . This shows that span ( E ) is dense in U α,β . Since span ( E ) is dense in U and span ( E ) is dense in U , span ( E α,β ) is dense in span ( E ) , and so span ( E α,β ) isdense in U α,β . Now let φ ∈ U ′ α,β . If FB ( φ ) = 0 , then < f, φ > = 0 for every f ∈ E α,β , andso φ = 0 since span ( E α,β ) is dense in U α,β , which shows that the Fourier-Boreltransform is one-to-one on U α,β . (cid:3) We will now give a way to compute < f, φ > for f ∈ V α,β ( X ) , φ ∈ U ′ α,β byusing Fourier-Borel transforms. For σ ∈ Π ≤ j ≤ k (cid:16) C \ S ∗ α j ,β j (cid:17) , define e ∗ σ ∈ U ′ α,β by using the formula < f, e ∗ σ > = FB ( f )( − σ ) . (27)Also for φ ∈ U ′ α,β , g ∈ U α,β , define φg ∈ U ′ α,β by using the formula < f, φg > = < f g, φ > ( f ∈ U α,β ) . It follows from definition 10.3 that if σ = ( σ , . . . , σ k ) ∈ Π ≤ j ≤ k (cid:16) C \ S ∗ α j ,β j (cid:17) , we have, for f ∈ U α,β , < f, e ∗ σ > = Z e iω . ∞ e σζ f ( ζ ) dζ, ω = ( ω , . . . , ω k ) satisfies α j ≤ ω j ≤ β j , Re ( σ j ω j ) < for ≤ j ≤ k, which gives k e ∗ σ k ∞ ≤ Π ≤ j ≤ k Z ∞ e tRe ( σ j ω j ) dt = 1Π ≤ j ≤ k ( − Re ( σ j ω j )) . The same formula as above holds with the same ω to compute < f, e ∗ σ ′ > for σ ′ ∈ Π ≤ j ≤ k (cid:16) C \ S ∗ α j ,β j (cid:17) when | σ − σ ′ | is sufficiently small, and so the map σ → e ∗ σ ∈ U ′ α,β is holomorphic on Π ≤ j ≤ k (cid:16) C \ S ∗ α j ,β j (cid:17) since the map λ → e − λ ∈ L ( R + ) is holomorphic on the open half-plane P + := { λ ∈ C | Re ( λ ) > } . Now let ǫ ∈ S ∗ α,β and let ω ∈ Π ≤ j ≤ k [ α j , β j ] such that Re ( ǫ j e iω j ) > for j ≤ k. Then σ − ǫ ∈ Π ≤ j ≤ k (cid:16) C \ S ∗ α j ,β j (cid:17) for σ ∈ ˜ ∂S ∗ α,β , and Re (( σ j − ǫ j ) ω j ) ≤− Re ( ǫ j ω j ) < for ≤ j ≤ k. We obtain k e ∗ σ − ǫ k ≤ ≤ j ≤ k Re ( ǫ j ω j ) , and so sup σ ∈ ˜ ∂S ∗ α,β k e ∗ σ − ǫ k ∞ < + ∞ ( ǫ ∈ S ∗ α,β ) . We now give the following certainly well-known natural result.
Proposition 10.8
Let φ ∈ U ′ α,β . Assume that Z ˜ ∂S ∗ α,β |FB ( φ )( σ ) || dσ | < + ∞ . Then we have, for ǫ ∈ S ∗ α,β ,φe − ǫ = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) e ∗ σ − ǫ dσ, where the Bochner integral is computed in ( U ′ α,β , k . k ∞ ) , which gives, for f ∈V α,β ( X ) , < f e − ǫ , φ > = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ + ǫ ) dσ. (28)Proof: Since the map σ → e ∗ σ − ǫ ∈ U ′ α,β is continuous on ˜ ∂S ∗ α,β , and since sup σ ∈ ˜ ∂S ∗ α,β k e ∗ σ − ǫ k ∞ < + ∞ , the Bochner integral R ˜ ∂S ∗ α,β FB ( φ )( σ ) e ∗ σ − ǫ dσ iswell-defined in ( U ′ α,β , k . k ∞ ) . Set φ ǫ := iπ ) k R ˜ ∂S ∗ α,β FB ( φ )( σ ) e ∗ σ − ǫ dσ ∈ U ′ α,β . Since the map φ → FB ( φ )( ζ ) is continuous on U ′ α,β , we have, for ζ ∈ S α,β , FB ( φ ǫ )( ζ ) = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( e ∗ σ − ǫ )( ζ ) dσ = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) < e − ζ , e ∗ σ − ǫ > dσ.
56t follows from definition (27) that < e − ζ , e ∗ σ − ǫ > = FB ( e − ζ )( ǫ − σ ) . Let ω ∈ Π ≤ j ≤ k [ α j , β j ] such that Re ( ǫ j e iω j ) > for j ≤ k. Since Re (( σ j − ǫ j ) ω j ) ≤− Re ( ǫ j ω j ) < for ≤ j ≤ k, we have, for σ ∈ ˜ ∂S ∗ α,β , FB ( e − ζ )( ǫ − σ ) = Z e iω . ∞ e ( σ − ǫ ) η e − ζ ( η ) dη = Z e iω . ∞ e ( σ − ǫ − ζ ) η dη = 1Π ≤ j ≤ k ( ζ j + ǫ j − σ j ) . Using the notation ζ + ǫ − σ := ≤ j ≤ k ( ζ j + ǫ j − σ j ) , this gives FB ( φ ǫ )( ζ ) = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) ζ + ǫ − σ dσ. As in appendix 3 , set W j,n ( ζ j ) = n n + e αj + βj i ζ j ! for n ≥ , ζ j ∈ S ∗ α j ,β j , andset W n ( ζ ) = Π j ≤ k W n,j ( ζ j ) for ζ ∈ S ∗ α,β . Then | W n,j ( ζ j ) | ≤ for ζ j ∈ S ∗ α j ,β j , W n ( ζ ) → as n → ∞ uniformly on compact sets of S ∗ α,β , and lim | ζ |→∞ ζ ∈ S ∗ α,β W n ( ζ ) =0 . The open set S ∗ α,β is admissible with respect to ( α, β ) in the sense of definition12.1 and, since FB ( φ ) is bounded on S ∗ α,β , FB ( φ ) W n ∈ H (1) ( S ∗ α,β ) for n ≥ . It follows then from theorem 12.5 that we have, for t ∈ (0 , , ζ ∈ S ∗ α,β , iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ + tǫ ) W n ( σ + tǫ ) ζ + (1 − t ) ǫ − σ dσ = 1(2 iπ ) k Z ˜ ∂S ∗ α,β + tǫ FB ( φ )( σ ) W n ( σ ) ζ + ǫ − σ dσ = FB ( φ )( ζ + ǫ ) W n ( ζ + ǫ ) , and it follows from the Lebesgue dominated convergence theorem that wehave iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) W n ( σ ) ζ + ǫ − σ dσ = FB ( φ )( ζ + ǫ ) W n ( ζ + ǫ ) . Taking the limit as n → + ∞ , and using again the Lebesgue dominatedconvergence theorem, we obtain, for ζ ∈ S ∗ α,β , FB ( φ ǫ )( ζ ) = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) ζ + ǫ − σ dσ = FB ( φ )( ζ + ǫ )= < e − ζ e − ǫ , φ > = < e − ζ , φe − ǫ > = FB ( φe − ǫ )( ζ ) , and it follows from the injectivity of the Fourier-Borel transform on U ′ α,β that φ ǫ = φe − ǫ . f ∈ V α,β ( X ) , since < f, e ∗ σ − ǫ > = FB ( f )( − σ + ǫ ) for σ ∈ ˜ ∂S ∗ α,β , < f e − ǫ , φ > = < f, φe − ǫ > = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) < f, e ∗ σ − ǫ > = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ + ǫ ) dσ. (cid:3) For J ⊂ { , . . . , k } , set P J,j = C \− S − π − α j , π − α j , ω J,i = α j for j ∈ J, P
J,j = C \ − S − π − β j , π − β j , ω J,i = β j for j ∈ { , . . . , k } \ J, and set P J = Π ≤ j ≤ k P J,i ,ω J = ( ω J, , . . . , ω J,k ) . If f ∈ V α,β ( X ) , and if R ˜ ∂S α,β k f ( ζ ) k X | dζ | < + ∞ , thenthe formula FB ( f )( σ ) = R e iωJ . ∞ e − ζσ f ( ζ ) dσ defines a continuous bounded ex-tension of FB ( f ) to P J . So in this situation FB ( f ) has a continuous boundedextension to ∪ J ⊂{ ,...,k } P J = Π ≤ j ≤ k (cid:16) C \ − S ∗ α j ,β j (cid:17) . Applying formula (28) tothe sequence ( ǫ n ) = ( ǫn ) for some ǫ ∈ S ∗ α,β , we deduce from the Lebesgue domi-nated convergence theorem and from formula (23) the following result. Corollary 10.9
Let f ∈ V α,β ( X ) , and let φ ∈ U ′ α,β . Assume that the followingconditions are satisfied ( i ) Z ˜ ∂S α,β k f ( ζ ) k X | dζ | < + ∞ . ( ii ) Z ˜ ∂S ∗ α,β |FB ( φ )( σ ) || dσ | < + ∞ . Then < f, φ > = 1(2 iπ ) k Z ˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ ) dσ. (29)In the following we will denote by ˜ ν the functional f → R S α,β f ( ζ ) dν ( ζ ) for ν ∈ M ( S α,β ) . In order to give a way to compute < f, φ > for φ ∈ U ′ α,β , f ∈V α,β ( X ) , we will use the following easy observation. Proposition 10.10
Let ν be a probability measure on S α,β , let R > , andlet X be a separable Banach space. Set ν R ( A ) = ν ( RA ) for every Borel set A ⊂ S α,β . Then lim R → + ∞ k f ˜ ν R − f k ∞ = 0 for every f ∈ U α,β ( X ) . Proof: Let f ∈ U α,β ( X ) . Then f is uniformly continuous on S α,β , and so forevery δ > there exists r > such that k f ( ζ + η ) − f ( ζ ) k X < δ for every ζ ∈ S α,β and for every η ∈ S α,β ∩ B (0 , r ) . It follows from the Lebesgue dominated conver-gence theorem that lim R → + ∞ ν R ( B (0 , r )) = lim R → + ∞ ν ( B (0 , rR )) = ν ( S α,β ) =1 . This gives 58 im sup R → + ∞ k f ˜ ν R − f k ∞ = lim sup R → + ∞ sup ζ ∈ S α,β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z S α,β ( f ( ζ + η ) − f ( ζ )) dν R ( η ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ! ≤ lim sup R → + ∞ sup ζ ∈ S α,β Z S α,β ∩ B (0 ,r ) k f ( ζ + η ) − f ( ζ ) k X dν n ( η ) ! +2 k f k ∞ lim sup R → + ∞ Z S α,β \ ( S α,β ∩ B (0 ,r )) dν n ( η ) ≤ δ. Hence lim R → + ∞ k f ˜ ν R − f k ∞ = 0 . (cid:3) It follows from the definition of ν R that < f, ˜ ν R > = < f R , ˜ ν > for f ∈V α,β ( X ) , where f R ( ζ ) = f ( R − ζ ) ( ζ ∈ S α,β ) . In particular if FB (˜ ν R ) = FB ( ν ) R , and (˜ ν ) R ∗ (˜ ν ) R = (˜ ν ∗ ˜ ν ) R = ( ^ ν ∗ ν ) R for R > if ν and ν are two probability measures on S α,β . We deduce from proposition 10.8 and proposition 10.9 the following corollary,in which the sequence ( W n ) n ≥ of functions on S ∗ α,β introduced in appendix3 and used in the proof of proposition 10.8 allows to compute < f, φ > for φ ∈ U ′ α,β , f ∈ V α,β ( X ) in the general case. Corollary 10.11
Set W n ( ζ ) = Π ≤ j ≤ k n n + ζ j e i αj + βj ! for n ≥ , ζ = ( ζ , . . . , ζ k ) ∈ S ∗ α,β . Then we have, for φ ∈ U ′ α,β , f ∈ V α,β ( X ) ,< f, φ > = lim ǫ → ǫ ∈ Sα,β lim n → + ∞ iπ ) k Z ˜ ∂S ∗ α,β W n ( σ ) FB ( φ )( σ ) FB ( f )( ǫ − σ ) dσ ! . (30)Proof: Define a measure ν on S α,β by using the formula < f, ν > = Z [0 , + ∞ ) k e − t ... − t k f ( t e i α β , . . . , t k e i αk + βk ) dt . . . dt k ( f ∈ C ( S α,β )) . Then ν and ν = ν ∗ ν are probability measures on S α,β , and we have, for ζ ∈ S ∗ α,β , FB (˜ ν )( ζ ) = Z [0 , + ∞ ) k e − t ... − t k e − t ζ e i α β − ... − t k ζ k e i αk + βk dt . . . dt k = Π ≤ j ≤ k
11 + ζ j e i αj + βj . FB (˜ ν ) = FB (˜ ν ) = W , and FB (˜ ν n ) = ( W ) n = W n . It follows from(29) that we have, for ǫ ∈ S ∗ α,β ,< f e − ǫ , φ > = lim n → + ∞ < ( f e − ǫ ) ˜ ν n , φ > = lim n → + ∞ < f e − ǫ , φ ∗ ˜ ν n > = lim n → + ∞ iπ ) k Z ˜ ∂S ∗ α,β FB ( φ ∗ ˜ ν n )( σ ) FB ( f )( ǫ − σ ) dσ = lim n → + ∞ iπ ) k Z ˜ ∂S ∗ α,β W n ( σ ) FB ( φ )( σ ) FB ( f )( ǫ − σ ) dσ, and the result follows from the fact that < f, φ > = lim ǫ → ǫ ∈ Sα,β < f e − ǫ , φ > . (cid:3)
11 Appendix 2: An algebra of fast-decreasingholomorphic functions on products of sectorsand half-lines and its dual
In this section we will use the notations introduced in definition 4.1 for α =( α , . . . , α k ) ∈ R k and β = ( β , . . . , β k ) ∈ R k ) satisfying α j ≤ β j < α j + π for ≤ j ≤ k. Notice that is x ∈ C , y ∈ C , there exists z ∈ C such that (cid:16) x + S ∗ α j ,β j (cid:17) ∩ (cid:16) y + S ∗ α j ,β j (cid:17) = z + S ∗ α j ,β j . Such a complex number z is uniqueif α j < β j . If α j = β j , then S ∗ α j ,β j = S α j − π/ ,α j + π/ is a closed half-plane,the family n x + S ∗ α j ,β j o x ∈ C is linearly ordered with respect to inclusion and thecondition (cid:16) x + S ∗ α j ,β j (cid:17) ∩ (cid:16) y + S ∗ α j ,β j (cid:17) = z + S ∗ α j ,β j defines a real line of theform z + e iα j R , where z ∈ { x, y } . The following partial preorder on C k is the partial order associated to thecone S ∗ α,β if α j < β j for ≤ j ≤ k. Definition 11.1 (i) For z = ( z , . . . , z k ) ∈ C k and z ′ = ( z ′ , . . . , z ′ k ) ∈ C k , set z (cid:22) z ′ if z ′ ∈ z + S ∗ α,β . (ii) if ( z ( j ) ) ≤ j ≤ m is a finite family of elements of C k denote by sup ≤ j ≤ m z j the set of all z ∈ C k such that ∩ ≤ j ≤ k (cid:16) z ( j ) + S ∗ α,β (cid:17) = z + S ∗ α,β . For z = ( z , . . . , z k ) ∈ C k , set e z = ( e z , . . . , e z k ) , and denote again by e z : C k → C the map ( ζ , . . . , ζ k ) → e zζ = e z ζ + ... + z k ζ k . It follows from (17) that e − z ′ U α,β ⊆ e − z U α,β if z (cid:22) z ′ . For f ∈ e − z V α,β , set k f k e − z V α,β = k e z f k ∞ , which defines a Banach spacenorm on e − z U α,β and e − z V α,β . Proposition 11.2 (i)Set γ n = ne − i α + β for n ≥ . Then the sequence ( γ n ) n ≥ is cofinal in ( C k , (cid:22) ) . ii) If z (cid:22) z ′ , then e − z ′ U α,β is a dense subset of ( e − z U α,β , k . k e − z U α,β ) . (iii) The set ∩ z ∈ C k e − z U α,β is a dense ideal of U α,β , which if a Fréchet algebrawith respect to the family ( k . k e − γn U α,β ) n ≥ . (iv) If X is a separable Banach space, and if z ∈ sup ≤ j ≤ m z ( j ) , then e − z U α,β ( X ) = ∩ ≤ j ≤ m e − z ( j ) U α,β ( X ) , e − z V α,β ( X ) = ∩ ≤ j ≤ m e − z ( j ) V α,β ( X ) , and k f k e − z V α,β =max ≤ j ≤ m k f k e − z ( j ) V α,β for f ∈ e − z V α,β . Proof: (i) Let z = ( z , . . . , z k ) ∈ C , and let j ≤ k. Since (cid:0) π − β j (cid:1) + (cid:0) − π − α j (cid:1) = − ( α j + β j ) , t ,j e − i ( αj + βj )2 ∈ ∂ ( z j + S ∗ α j ,β j ) for some t ,j ∈ R , so te − i ( αj + βj )2 ∈ z j + S ∗ α j ,β j for every t ≥ t ,j , and (i) follows.(ii) Assume that z (cid:22) z ′ . The fact that e − z ′ U α,β ⊂ e − z U α,β follows from (16).Let z ′′ ∈ z ′ + S ∗ α,β ⊂ z + S ∗ α,β . We have z ′′ = z + re iη where r > , and where η = ( η , . . . , η k ) satisfies − π − α j < η j < π − β j for j ≤ k. The semigroup ( e − te iη ) t> is analytic and bounded in the Banach algebra U α,β , and lim t → + k f − f e − te iη k ∞ = 0 for every f ∈ U α,β . It follows then from the analyticity of this semigroup that [ e − re iη U α,β ] − =[ ∪ t> e − te iη U α,β ] − = U α,β . Hence e − z ′′ U α,β is dense in e − z U α,β , which proves(ii) since e − z ′′ U α,β ⊂ e − z ′ U α,β . (iii) Denote by i z,z ′ : f → f the inclusion map from e − z ′ U α,β into e − z U α,β for z (cid:22) z ′ . Equipped with these maps, the family ( e − z U α,β ) z ∈ C k is a projectivesystem of Banach spaces, and we can identify (cid:0) ∩ z ∈ C k e − z U α,β , ( k . k e z U α,β ) z ∈ C k (cid:1) to the inverse limit of this system, which defines a structure of complete locallyconvex topological space on (cid:0) ∩ z ∈ C e − z U α,β , ( k . k e − z U α,β ) z ∈ C k (cid:1) . It follows from(i) that the sequence ( k . k e − γn U α,β ) n ≥ of norms defines the same topology asthe family (cid:0) k . k e − z U α,β ) z ∈ C (cid:1) on ∩ z ∈ C e − z U α,β = ∩ n ≥ e − γ n U α,β , which defines aFréchet algebra structure on ∩ z ∈ C e − z U α,β .It follows from (ii) that e − γ n +1 U α,β is dense in e − γ n U α,β for n ≥ , and a stan-dard application of the Mittag-Leffler theorem of projective limits of completemetric spaces, see for example theorem 2.14 of [13], shows that ∩ z ∈ C e − z U α,β = ∩ n ≥ e − γ n U α,β is dense in e − γ U α,β = U α,β . (iv) Let z = ( z , . . . , z k ) ∈ C k , let z ′ = ( z ′ , . . . z ′ k ) ∈ C k , and let z ” =( z ” , . . . , z ” k ) ∈ sup ( z, z ′ ) . Then e − z ”+ z ∈ V α,β , e − z ”+ z ′ ∈ V α,β , k e − z ”+ z k ∞ ≤ , k e − z ”+ z k ∞ ≤ , and so e − z ” U α,β ( X ) ⊂ e − z U α,β ( X ) ∩ e − z ′ U α,β ( X ) , e − z ” V α,β ( X ) ⊂ e − z V α,β ( X ) ∩ e − z ′ V α,β ( X ) , and max( k f k e − z V α,β ( X ) , k f k e − z ′ V α,β ( X ) ) ≤ k f k e − z ” V α,β ( X ) for f ∈ e − z ” V α,β ( X ) . We claim that | e z j ” ζ | ≤ min( | e z j ζ | , | e z ′ j ζ | ) for ζ ∈ ∂S α j ,β j , ≤ j ≤ k. If z j ∈ z ′ j + S ∗ α j ,β j , or if z ′ j ∈ z j + S ∗ α j ,β j , this is obviously true.Otherwise we have α j < β j and, say, z j ” = z j + re i ( − α j − π ) = z ′ j + r ′ e i ( − β j + π ) , with r > , r ′ > . Let ζ = ρe iθ ∈ S α j ,β j , where ρ ≥ , θ ∈ [ α j , β j ] . We have Re (( z ” j − z j ) ζ ) = rρcos ( θ − α j − π ) ≥ , and Re (( z ” j − z j ) ζ ) = r ′ ρcos ( θ − β j + π ) ≥ . So | e z j ” ζ | = | e zζ | ≤ | e z ′ ζ | if θ = α j , and | e z j ” ζ | = | e z ′ ζ | ≤ | e z ′ ζ | if θ = β j , which proves the claim.We now use the Phragmén-Lindel ¨ of principle. Let s ∈ ∪ ≤ j ≤ k (1 , πβ j − α j ) and for ≤ j ≤ k let ζ sj be a continuous determination of the s -power of ζ S α j ,β j which is holomorphic on S α j ,β j when α j < β j . Set ζ s = ζ s . . . ζ sk for ζ ∈ S α,β . Let f ∈ e − z V α,β ( X ) ∩ e − z ′ V α,β ( X ) , and let ǫ > . Set g ǫ ( s ) = e − ǫζ s e z ” ζ f ( ζ ) for ζ ∈ S α,β . It follows from the claim that g ǫ ∈ U α,β ( X ) , and itfollows from the maximum modulus principle that there exists ζ ∈ ˜ ∂S α,β suchthat k g ǫ k U α,β ( X ) = k g ǫ ( ζ ) k ≤ | e − ǫζ s | max( k f k e − z V α,β ( X ) , k f k e − z ′ V α,β ( X ) ) ≤ max( k f k e − z V α,β ( X ) , k f k e − z ′ V α,β ( X ) ) . Since lim ǫ → e − ǫζ s = 1 for every ζ ∈ S α,β , this shows that f ∈ e − z ” V α,β ( X ) , and k f k e − z ” V α,β ( X ) = max( k f k e − z V α,β ( X ) , k f k e − z ′ V α,β ( X ) ) . Now let f ∈ e − z U α,β ( X ) ∩ e − z ′ U α,β ( X ) . Then f ∈ e − z ” V α,β , and lim | ζ |→ ζ ∈ ∂Sα,β k e z ” ζ f ( ζ ) k =0 . The Banach algebra U α,β possesses a bounded approximate identity ( g n ) n ≥ , one can take for example g n ( ζ , . . . , ζ n ) = Π ≤ j ≤ k nζ j nζ j + e i αj + βj . We have lim n → + ∞ k e z ” f g n − e z ” f k ∞ = lim n → + ∞ max ζ ∈ ∂S α,β k e z ” ζ f ( ζ ) g n ( ζ ) − e z ” ζ f ( ζ ) k = 0 , and so e z ” f ∈ U α,β since U α,β is a closed subalgebra of V α,β . This concludes theproof of (iv) when m = 2 . The general case follows by an immediate induction,since sup( ζ, z ( l ) ) = sup ≤ j ≤ l z ( j ) ) for every ζ ∈ sup ≤ j ≤ l − z ( j ) if ( z , . . . , z ( l ) ) is a finite family of elements of C k . (cid:3) Notice that assertions (ii) and (iii) of the proposition do not extend to thecase where β j = α j + π for some j ≤ k. It suffices to consider the case where α j = − π , β j = π . Set λ j ( t ) = ( λ s,t ) ≤ s ≤ k , where λ s,t = 0 for s = j and λ j,t = t. Then the map f → e − λ j ( t ) f is an isometry on U α,β for every t ≥ and ∩ t> u λ j ( t ) U α,β = { } since the zero function is the only bounded holomorphicfunction f on the right-hand open half-plane satisfying lim r → + ∞ | e tr f ( r ) | = 0 for every t > . Let i ζ : f → f be the inclusion map from ∩ z ∈ C e − z U α,β into e − ζ U α,β . Since i ζ has dense range, the map i ∗ ζ : φ → φ | ∩ z ∈ C e − z U α,β is a one-to-one map from ( e − ζ U α,β ) ′ into ∩ z ∈ C e − z U α,β ) ′ , which allows to identify ( e − ζ U α,β ) ′ to a subsetof ( ∩ z ∈ C e − z U α,β ) ′ , so that we have ( ∩ z ∈ C e − z U α,β ) ′ = ∪ z ∈ C ( e − z U α,β ) ′ = ∪ n ≥ ( e − ne − i α + β U α,β ) ′ . (31) Definition 11.3
Set F α,β := ( ∩ z ∈ C k e − z U α,β ) ′ . Let φ ∈ F α,β , and let X be aseparable Banach space.(i) The domain of the Fourier-Borel transform of φ is defined by the formula Dom ( FB ( φ )) := { z ∈ C k | φ ∈ ( e − z U α,β ) ′ } . (ii) For z ∈ Dom ( FB ( φ )) the functional φe − z ∈ U ′ α,β is defined by theformula < f, φe − z > = < e − z f, φ > ( f ∈ U α,β ) , nd < g, φ > is defined for g ∈ e − z V α,β ( X ) by the formula < g, φ > = < e z g, φe − z > . (iii) The Fourier-Borel transform of φ is defined for z ∈ Dom ( FB ( φ )) bythe formula FB ( φ )( z ) = < e − z , φ > . (iv) The z -Cauchy transform of φ is defined on C k \− S ∗ α,β for z ∈ Dom ( FB ( φ )) by the formula C z ( φ ) = C ( φe − z ) . (v) If z ∈ Dom ( FB ( φ )) a measure ν of bounded variation on S α,β is said tobe a z -representing measure for φ if ν is a representing measure for φe − z . Since the map ζ → e − ζ is holomorphic on S ∗ α,β , the map z → e − z is aholomorphic map from λ + S ∗ α,β into e − λ U α,β for every λ ∈ Dom ( FB ( φ )) , and so FB ( φ ) is holomorphic on the interior of Dom ( FB ( φ )) for φ ∈ F α,β . Also the z -Cauchy transform C z ( φ ) is holomorphic on C \ S α,β for every z ∈ Dom ( FB ( φ )) . Notice also that if φ ∈ U ′ α,β , then S ∗ α,β ⊂ Dom ( FB ( φ )) andso the function FB ( φ ) defined above is an extension to Dom ( FB ( φ )) of theFourier-Borel transform already introduced in definition 10.3 on S ∗ α,β . Now let z ∈ Dom ( FB ( φ )) , z ′ ∈ Dom ( FB ( φ )) and assume that g ∈ e − z V α,β ( X ) ∩ e − z ′ V α,β ( X ) . Let z ′′ ∈ sup( z, z ′ ) ⊂ Dom ( FB ( φ )) . Then g ∈ e − z ” V α,β ( X ) . Let ν be a z -representative measure for φ. We have, for h ∈ ∩ λ ∈ C k e − λ U α,β , since e − z = e − z ′′ e z ′′ − z .< h, φ > = Z S α,β e z ( ζ ) h ( ζ ) dν ( ζ ) = Z S α,β e z ′′ ( ζ ) h ( ζ ) e z − z ′′ ( ζ ) dν ( ζ ) . Since e z − z ′′ ν is a measure of bounded variation on S α,β , e z − z ” ν is a z ′′ -representative measure for φ. Similarly if ν ′ is a z ′ -representative measure for φ then e z ′ − z ′′ ν ′ is a z ′′ -representative measure for φ, and we have Z S α,β e z ( ζ ) g ( ζ ) dν ( ζ ) = Z S α,β e z ′′ ( ζ ) g ( ζ ) e z − z ′′ ( ζ ) dν ( ζ )= Z S α,β e z ′′ ( ζ ) g ( ζ ) e z ′ − z ′′ ( ζ ) dν ′ ( ζ ) = Z S α,β e z ′ ( ζ ) g ( ζ ) dν ′ ( ζ ) , which shows that the definition of < g, φ > does not depend on the choiceof z ∈ Dom ( FB ( φ )) such that g ∈ e − z V α,β ( X ) . Proposition 11.4
Let φ ∈ F α,β . (i) The set Dom ( FB ( φ )) is connected.(ii) z + S ∗ α,β ⊂ Dom ( FB ( φ )) , and FB ( φ ) is continuous on z + S ∗ α,β andholomorphic on z + S ∗ α,β for every z ∈ Dom ( FB ( φ )) . Dom ( FB ( φ )) is connected follows from the fact thatthe arcwise connected set ( z + S ∗ α,β ) ∪ ( z + S ∗ α,β ) is contained in Dom ( FB ( φ )) for z ∈ Dom ( FB ( φ )) , z ∈ Dom ( FB ( φ )) . (ii) Let z ∈ Dom ( FB ( φ )) . It follows from (16) that z + S ∗ α,β ⊂ Dom ( FB ( φ )) and so FB ( φ ) is holomorphic on the open set z + S ∗ α,β ⊂ Dom ( FB ( φ )) . Let ν be a measure of bounded variation on S α,β which is z -representing measure for φ. We have, for η ∈ S α,β , FB ( φ )( z + η ) = < e − z − η , φ > = < e − η , φe − z > Z S α,β e − ηζ dν ( ζ ) , and the continuity of FB ( φ ) on z + S ∗ α,β follows from the Lebesgue dominatedconvergence theorem. (cid:3) Notice that
Dom ( FB ( φ )) is not closed in general: for example if we set < f, φ > = R S − π , π ζf ( ζ ) dm ( ζ ) for f ∈ ∩ z ∈ C e − z U − π , π , where m denotes theLebesgue measure on C , then t ∈ Dom ( FB ( φ )) for every t > , but / ∈ Dom ( FB ( φ )) . Notice also that if ν is a measure supported by a compact sub-set of S α,β , and if we set < f, φ > := R S α,β f ( ζ ) dν ( ζ ) for f ∈ ∩ z ∈ C k e − z U α,β , then φ ∈ ∩ z ∈ C k ( e − z U α,β ) ′ , so that Dom ( FB ( φ )) = C k , and FB ( φ ) is the entirefunction defined on C k by the formula FB ( φ )( z ) = Z S α,β e − zζ dν ( ζ ) . We now introduce the convolution product of elements of F α,β . If φ ∈F α,β , f ∈ ∩ z ∈ C k e − z U α,β , λ ∈ S α,β , set again f λ ( ζ ) = f ( ζ + λ ) for ζ ∈ S α,β . Then f λ ∈ ∩ z k ∈ C e − z U α,β , and we can compute < f λ , φ > . The map λ → f λ is a continuous map from S α,β into the Fréchet algebra ∩ z ∈ C e − z U α,β which isholomorphic on S α,β . We obtain
Lemma 11.5
Let φ ∈ F α,β . Then the function f φ : λ → < f λ , φ > belongsto ∩ z ∈ C k e − z U α,β for every f in ∩ z ∈ C k e − z U α,β , and the linear map f → f φ iscontinuous on ∩ z ∈ C k e − z U α,β . Proof: Let f ∈ ∩ z ∈ C k e − z U α,β , let z ∈ Dom ( FB ( φ )) , let ν be a z -representingmeasure for φ on S α,β , and let z ∈ C k . Let z ∈ sup( z , z ) , so that ( z + S ∗ α,β ) ∩ ( z + S ∗ α,β ) = z + S ∗ α,β , and set η = z − z , η = z − z. We have, for λ ∈ S α,β ,e zλ < f λ , φ > = Z S α,β e zλ + z ζ f ( ζ + λ ) dν ( ζ ) = Z S α,β e − ηλ − η ζ e z ( ζ + λ ) f ( ζ + λ ) dν ( ζ ) . Since | e − ηλ − η ζ e z ( ζ + λ ) f ( ζ + λ ) | ≤ k e z f k ∞ , it follows from Lebesgue’sdominated convergence theorem that lim | λ |→ + ∞ λ ∈ Sα,β | e zλ < f λ , φ > | = 0 , and so64 φ ∈ ∩ z ∈ C k e − z U α,β . Also k e z f φ k ∞ ≤ k e z f k ∞ R S α,β d | ν | ( ζ ) , which shows thatthe map f → f φ is continuous on ∩ z k ∈ C e − z U α,β . (cid:3) Notice that it follows from the Hahn-Banach theorem that given φ ∈ ( e − z U α,β ) ′ there exists a z -representing measure ν for φ such that R S α,β d | ν | ( ζ ) = k φ k ( e − z U α,β ) ′ . The calculation above shows then that we have, for z ∈ C , f ∈ ∩ z ∈ C e − z U α,β ,φ ∈ F α,β , z ∈ Dom ( FB ( φ )) , z ∈ sup( z , z ) , k e z f φ k ∞ ≤ k e z f k ∞ k φ k ( e − z U α,β ) ′ . (32) Proposition 11.6
For φ ∈ F α,β , φ ∈ F α,β , define the convolution product φ ∗ φ ∈ F α,β by the formula < f, φ ∗ φ > = < f φ , φ > ( f ∈ ∩ z ∈ C k e − z U α,β ) . Then sup ( z , z ) ⊂ Dom ( FB ( φ ∗ φ )) for z ∈ Dom ( FB ( φ )) , z ∈ Dom ( FB ( φ )) , and we have, for z ∈ sup( z , z ) , k φ ∗ φ k ( e − z U α,β ) ′ ≤ k φ k ( e − z U α,β ) ′ k φ k ( e − z U α,β ) ′ . More generally
Dom ( FB ( φ )) ∩ Dom ( FB ( φ )) ⊂ Dom ( FB ( φ ∗ φ )) , andif z ∈ Dom ( FB ( φ )) ∩ Dom ( FB ( φ )) then ( φ ∗ φ ) e − z = ( φ e − z ) ∗ ( φ e − z ) , so that ν ∗ ν is a z -representative measure for φ ∗ φ if ν is a z -representingmeasure for φ and if ν is a z -representing measure for ν , and we have FB ( φ ∗ φ )( z ) = FB ( φ )( z ) FB ( φ )( z ) ( z ∈ Dom ( FB ( φ )) ∩ Dom ( FB ( φ )) . Proof: Let z ∈ Dom ( FB ( φ )) , let z ∈ Dom ( FB ( φ )) , and let z ∈ sup( z , z ) . It follows from (32) that we have, for f ∈ ∩ z ∈ C k e z U α,β , | < f, φ ∗ φ > | = | < f φ , φ > | ≤ k e z f φ k ∞ k φ k ( e − z U α,β ) ′ ≤ k e z f k ∞ k φ k ( e − z U α,β ) ′ k φ k ( e − z U α,β ) ′ . Hence φ ∗ φ ∈ F α,β , sup ( z , z ) ⊂ Dom ( FB ( φ ∗ φ )) , and k φ ∗ φ k ( e − z U α,β ) ′ ≤ k φ k ( e − z U α,β ) ′ k φ k ( e − z U α,β ) ′ for z ∈ sup( z , z ) . Let z ∈ Dom ( FB ( φ )) ∩ FB ( φ )) . Then z ∈ sup( z, z ) ⊂ Dom ( FB ( φ ∗ φ )) . Let ν be a z -representing measure for φ and let ν be a z -representingmeasure for φ . We have, for f ∈ ∩ s ∈ C k e − s U α,β ,< f, φ ∗ φ > = < f φ , φ > = Z S α,β e zλ f φ ( λ ) dν ( λ )= Z S α,β "Z S α,β e zζ f ( ζ + λ ) dν ( λ ) e zλ dν ( λ )= Z Z S α,β × S α,β e z ( ζ + λ ) f ( ζ + λ ) dν ( ζ ) dν ( λ ) = Z S α,β e zs d ( ν ∗ ν )( s ) , ν ∗ ν is a representing measure for ( φ ∗ φ ) e − z , which means that ν ∗ ν is a z -representative measure for φ ∗ φ . Since ν is a representativemeasure for φ e − z , and since φ is a representative measure for φ e − z , it followsfrom proposition 10.5 (ii) that ( φ ∗ φ ) e − z = ( φ e − z ) ∗ ( φ e − z ) . It follows also from proposition 10.5(ii) that FB ( φ ∗ φ )( z ) = FB (( φ ∗ φ ) e − z )(1) = FB (( φ e − z ) ∗ ( φ e − z ))(1)= FB ( φ e − z )(1) FB ( φ e − z )(1) = FB ( φ )( z ) FB ( φ )( z ) . (cid:3) Using proposition 10.4, we obtain the following link between z -Cauchy trans-forms and Fourier-Borel transforms of elements of F α,β . Proposition 11.7
Let φ ∈ F α,β . For j ≤ k, set I η,j = ( π − η, π − β j ] for η ∈ ( β j , α j + π ] , I η,j = ( − π − α j , π − β j ) for η ∈ ( α j + π, β j + π ] , and set I η = ( − π − α j , π − η ) for η ∈ ( β j + π, α j + 2 π ) . Then I η,j ⊂ [ − π − α j , π − β j ] ,cos ( η + s ) < for s ∈ I η,j , and if λ = ( λ , . . . , λ k ) ∈ C k \ S α,β , we have for ω = ( ω , . . . , ω k ) ∈ Π ≤ j ≤ k I arg ( λ j ) ,j , z ∈ Dom ( FB ( φ )) , C z ( φ )( λ ) = 1(2 iπ ) k Z e iω . ∞ e λσ FB ( φ )( σ + z ) dσ := 1(2 πi ) k Z e iω . ∞ . . . Z e iωk . ∞ e λσ FB ( φ )( σ + z ) dσ. (33)Proof: We have C z ( φ ) = C ( φe − z ) , and, for σ ∈ S ∗ α,β , FB ( φ )( σ + z ) = < e − σ − z , φ > = < e − σ e − z , φ > = < e − σ , φe − z > = FB ( φe − z )( σ ) . Applying formula (22) to φe − z , we obtain (33). (cid:3) Let X be a separable Banach space. For η ∈ S α,β , z ∈ C k , f ∈ e − z V α,β ( X ) , set f η ( ζ ) = f ( ζ + η ) ( ζ ∈ S α,β ) . If φ ∈ F α,β , and if z ∈ Dom ( FB ( φ )) , we have < f, φ ∗ δ η > = < e z f, ( φ ∗ δ η ) e − z > = < e z f, ( φe − z ) ∗ ( δ η e − z ) > = e − zη < e z f, ( φe − z ) ∗ δ η > = e − zη < ( e z f ) η , ( φe − z ) > = < e z f η , φe − z > = < f η , φ > . We also have, for f ∈ e − z U α,β ( X ) , lim η → η ∈ Sα,β k f η − f k e − z U α,β ( X ) = lim η → η ∈ Sα,β sup ζ ∈ S α,β k e zζ f ( ζ + η ) − e zζ f ( ζ ) k ∞ ≤ lim η → η ∈ Sα,β (cid:0) k ( e z f ) η − e z f k ∞ + | − e − zη |k ( e z f ) η ) k ∞ (cid:1) = 0 , ( e − ǫ f ) η = e − ǫη e − ǫ f η , lim η → ,η ∈ Sα,βǫ → ,ǫ ∈ S ∗ α,β k ( e − ǫ f ) η − f k e − z U α,β = lim η → ,η ∈ Sα,βǫ → ,ǫ ∈ S ∗ α,β k e − ǫ f η − f k e − z U α,β = 0 ( f ∈ e − z U α,β ( X ) , z ∈ C k ) (34)Now let f ∈ e − z V α,β ( X ) , and let φ ∈ ( e − z U α,β ) ′ . If ν is a z -representativemeasure for φ, we have, for η ∈ S α,β , ǫ ∈ S ∗ α,β ,< ( e − ǫ f ) η , φ > = e − ǫη < e − ǫ f η , φ > = e − ( ǫ + z ) η Z S α,β e − ǫζ e z ( ζ + η ) f ( ζ + η ) dν ( ζ ) , and it follows from the Lebesgue dominated convergence theorem that wehave lim η → ,η ∈ Sα,βǫ → ,ǫ ∈ S ∗ α,β k < ( e − ǫ f ) η , φ > − < f, φ > k X = lim η → ,η ∈ Sα,βǫ → ,ǫ ∈ S ∗ α,β k < e − ǫ f η , φ > − < f, φ > k X = 0 ( f ∈ e − z V α,β ( X ) , φ ∈ ( e − z U α,β ) ′ , z ∈ C k ) (35)The following consequence of proposition 10.6 allow to compute in some cases < f, φ > for φ ∈ U ′ α,β , f ∈ V α,β ( X ) , z ∈ Dom ( FB ( φ )) by using the z -Cauchytransform. Proposition 11.8
Assume that α j < β j < α j + π for ≤ j ≤ k, let φ ∈ F α,β , let z ∈ Dom ( FB ( φ )) , and let X be a separable Banach space.If f ∈ e − z V α,β ( X ) , and if Z ˜ ∂S α,β e Re ( zσ ) k f ( σ ) k X | dσ | < + ∞ , then we have, for η ∈ S α,β ,< f η , φ > = < f, φ ∗ δ η > = Z ˜ ∂S α,β e z ( σ − η ) C z ( φ )( σ − η ) f ( σ ) dσ. (36) In particular we have, for f ∈ V α,β ( X ) , ǫ ∈ S ∗ α,β , η ∈ S α,β ,e − ǫη < e − ǫ f η , φ > = < e − ǫ f, φ ∗ δ η > = Z ˜ ∂S α,β e ( z − ǫ )( σ − η ) C z ( φ )( σ − η ) f ( σ ) dσ. (37)67roof: Assume that f ∈ e − z V α,β ( X ) satisfies the condition R ˜ ∂S α,β k f ( σ ) k X | dσ | < + ∞ . We have, for η ∈ S α,β , ǫ ∈ S ∗ α,β ,< f η , φ > = < e z f η , φ e − z > = e − zη < ( e z f ) η , φe − z >, e − ǫ f η = e ǫη ( e − ǫ f ) η so (36) follows from (25) applied to e z f and φe − z , and (37) follows from (36)applied to e − ǫ f. (cid:3) For z ∈ C k , f ∈ e − z H ∞ ( S α,β , X ) , define the Fourier-Borel transform of f for ζ = ( ζ . . . , ζ k ) ∈ Π ≤ j ≤ k (cid:16) C \ ( − z j − S ∗ α j ,β j ) (cid:17) by the formula FB ( f )( ζ ) = FB ( e z f )( z + ζ ) = Z e iω . ∞ e − ζσ f ( σ ) dσ := Z e iω . ∞ . . . Z e iωk . ∞ e − ζ σ ... − ζ k σ k f ( σ , . . . , σ k ) dσ . . . dσ k , (38)where α j ≤ ω j ≤ β j and where Re (( z j + ζ j ) e iω j ) > for ≤ j ≤ k. The following consequences of proposition 10.8, corollary 10.9 and corollary10.11 allow to interpret the action of φ ∈ F α,β on e − z U α,β for z ∈ Dom ( FB ( φ )) in terms of Fourier-Borel transforms. Proposition 11.9
Let φ ∈ F α,β , let z = ( z , . . . , z k ) ∈ Dom ( FB ( φ )) , and let f ∈ e − z V α,β ( X ) . Set again W n ( ζ ) = Π ≤ j ≤ k n n + ζ j e i αj + βj ! for ζ = ( ζ , . . . , ζ k ) ∈ S ∗ α,β , n ≥ . Then ( i ) < f, φ > = lim ǫ → ǫ ∈ Sα,β lim n → + ∞ iπ ) k Z z +˜ ∂S ∗ α,β W n ( σ − z ) FB ( φ )( σ ) FB ( f )( − σ + ǫ ) dσ ! . (ii) If, further, R ˜ ∂S ∗ α,β |FB ( φ )( σ )) || dσ | < + ∞ , then we have, for ǫ ∈ S ∗ α,β ,< e − ǫ f, φ > = 1(2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ + ǫ ) dσ, and so < f, φ > = lim ǫ → ǫ ∈ Sα,β iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ + ǫ ) dσ. If, further, R ˜ ∂S α,β e Re ( zσ ) k f ( σ ) k| dσ | < + ∞ , then(iii) < f, φ > = 1(2 iπ ) k Z z +˜ ∂S ∗ α,β FB ( φ )( σ ) FB ( f )( − σ ) dσ. < f, φ > = < e z f, φe − z > . Since < FB ( f )( − ζ − z ) = FB ( e z f )( − ζ ) for ζ ∈ Π ≤ j ≤ k (cid:16) C \ S ∗ α,β (cid:17) , and since FB ( φe − z )( ζ ) = < e − ζ − z , φ > = FB ( φ )( ζ + z ) for ζ ∈ S ∗ α,β , it follows from corollary 10.11 that we have < f, φ > = < e z f, φe − z > = lim ǫ → ǫ ∈ S ∗ α,β iπ ) k Z ˜ ∂S ∗ α,β W n ( ζ ) FB ( φe − z )( ζ ) FB ( e z f )( ǫ − ζ ) dζ ! = lim ǫ → ǫ ∈ S ∗ α,β iπ ) k Z ˜ ∂S ∗ α,β W n ( ζ ) FB ( φ )( z + ζ ) FB ( f )( − z + ǫ − ζ ) dζ ! , and we obtain (i) by using the change of variables σ = z + ζ for ζ ∈ ˜ ∂S ∗ α,β . Usingthe same change of variables we deduce (ii) from proposition 10.8 and (iii) fromcorollary 10.9. (cid:3)
Lemma 11.10
Let α = ( α , . . . , α k ) ∈ R k , α ′ = ( α ′ , . . . , α ′ k ) ∈ R k , and assumethat α ′ j ≤ α j ≤ β j ≤ β ′ j < α ′ j + π for j ≤ k. Then ∩ z ∈ C k e − z U α ′ ,β ′ is dense in ∩ z ∈ C k e − z U α,β . Proof: Let φ ∈ F α,β , and assume that < f, φ > = 0 for every f ∈ ∩ z ∈ C k e − z U α ′ ,β ′ . Let z ∈ Dom ( FB ( φ )) . Then FB ( φ )( z + ζ ) = 0 for every ζ ∈ S ∗ α ′ ,β ′ . Since
Dom ( FB ( φ )) is connected, we have FB ( φ ) = 0 . Hence φ = 0 , since the Fourier-Borel transform is one-to-one on F α,β . (cid:3) We can thus identify F α,β to a subset of F α ′ ,β ′ for α ′ j ≤ α j ≤ β j ≤ β ′ j <α ′ j + π for j ≤ k. A standard application of the Mittag-Leffler theorem of projective limits ofcomplete metric spaces, see for example [13], theorem 2.14, shows that we havethe following result, where as before M a,b = { ( α, β ) ∈ R k × R k | a j < α j ≤ β j
Let a = ( a , . . . , a k ) ∈ R k , b = ( b , . . . , b k ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k. Then ∩ ( α ′ ,β ′ ) ∈ M a,b ,λ ∈ C k e − λ U α ′ ,β ′ is dense in e − z U α,β for every z ∈ C and every ( α, β ) ∈ M a,b . Let ( a, b ) ∈ R k × R k be as above, and denote by ∆ a,b the set of all triples ( α, β, z ) where ( α, β ) ∈ M a,b and z ∈ C k . Denote by (cid:22) the product partial order on R k associated to the usual order on R . If ( α, β, z ) ∈ ∆ a,b , ( α ′ , β ′ , z ′ ) ∈ ∆ a,b , set ( α, β, z ) (cid:22) ( α ′ , β ′ , z ′ ) if α ′ (cid:22) α, β (cid:22) β ′ and z ′ ∈ z + S ∗ α ′ ,β ′ . For every finitefamily F = { ( α ( l ) , β ( l ) , z ( l ) ) } ≤ l ≤ m of elements of ∆ a,b , set sup( F ) = { inf ≤ l ≤ m α ( l ) } × { sup ≤ l ≤ m β ( l ) } × sup ≤ l ≤ m z ( j ) , where sup ≤ l ≤ m z ( j ) denotes the set of all z ∈ C k satifsfying the condition z + S ∗ inf ≤ l ≤ m α ( l ) , sup ≤ l ≤ m β ( l ) = ∩ ≤ l ≤ m (cid:16) z ( l ) + S ∗ inf ≤ l ≤ k α ( l ) , sup ≤ l ≤ k β ( l ) (cid:17) ,
69o that sup ≤ l ≤ m z ( j ) , is the set introduced in definition 9.1(ii) when α =inf ≤ l ≤ m α ( l ) and β = sup ≤ l ≤ m β ( l ) . Notice that sup ≤ l ≤ m z ( j ) , is a singleton if (inf ≤ l ≤ m α ( l ) ) j < (sup ≤ l ≤ m β ( l ) ) j for ≤ j ≤ k. It follows from the proposition that we can identify the dual of the projectivelimit ∩ ( α,β,z ) ∈ ∆ a,b e − z U α,β to the inductive limit ∪ ( α,β,z ) ∈ ∆ a,b ( e − z U α,β ) ′ . Thissuggests the following definition.
Definition 11.12
Let a = ( a , . . . , a k ) ∈ R k , b = ( b , . . . , b k ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k. Set G a,b = ( ∩ ( α,β,z ) ∈ ∆ a,b e − z U α,β ) ′ = ∪ ( α,β,z ) ∈ ∆ a,b ( e − z U α,β ) ′ . For φ ∈ G a,b , set dom ( φ ) = { ( α, β, z ) ∈ ∆ α,β | φ ∈ ( e − z U α,β ) ′ } . We thus see that the inductive limit G a,b = ∪ ( α,β ) ∈ M a,b F α,β is an associativeunital pseudo-Banach algebra with respect to the convolution product intro-duced above on the spaces F α,β . A subset V of G a,b is bounded if and onlyif there exists ( α, β ) ∈ M a,b and z ∈ C k such that V is a bounded subset of ( e − z U α,β ) ′ . The proof of the following proposition is left to the reader.
Proposition 11.13
Let φ ∈ G a,b , and let ( α, β, z ) ∈ dom ( φ ) . Then ( α ′ , β ′ , z ′ ) ∈ dom ( φ ) if ( α, β, z ) (cid:22) ( α ′ , β ′ , z ′ ) . In particular if ( φ j ) ≤ j ≤ m is a finite familyof elements of G a,b , and if ( α ( j ) , β ( j ) , z ( j ) ) ∈ dom ( φ j ) for ≤ j ≤ m, then sup ≤ j ≤ m ( α ( j ) , β ( j ) , z ( j ) ) ⊂ ∩ ≤ j ≤ m dom ( φ j ) ⊂ dom ( φ ∗ . . . ∗ φ m ) .
12 Appendix 3: Holomorphic functions on ad-missible open sets
Definition 12.1
Let a = ( a , . . . , a p ) ∈ R k , b = ( b , . . . , b p ) ∈ R k such that a j ≤ b j ≤ a j + π for j ≤ k. An open set U ⊂ C k is said to be admissible with respect to ( α, β ) ∈ M a,b if U = Π ≤ j ≤ k U j , where the open sets U j ⊂ C satisfy the following conditions forsome z = ( z , . . . , z k ) ∈ C k , (i) U j + S ∗ α j ,β j ⊂ U j (ii) U j ⊂ z j + S ∗ α j ,β j , and ∂U j − z j = ( e ( − π − α j ) i . ∞ , e ( − α j − π ) i s ,j ) ∪ θ j ([0 , ∪ ( e ( π − β j ) i s ,j , e ( π − β j ) i . ∞ ) , where s ,j ≥ , s ,j ≥ , and where θ j : [0 , → S ∗ α j ,β j \ (cid:0) e ( − π − α j ) i . ∞ , e ( − α j − π ) i s j, ) ∪ ( e ( π − β j ) i s j, , e ( π − β j ) i . ∞ ) (cid:1) is a one-to-one piecewise- C curve such that θ j (0) = e ( − α j − π ) i s j, , and θ j (1) = e ( π − β j ) i s j, . If U is an admissible open set with respect to some ( α, β ) ∈ M a,b , H (1) ( U ) denotes the space of all functions F holomorphic on U such that k F k H (1) ( U ) :=sup ǫ ∈ S ∗ α,β R ˜ ∂U + ǫ | F ( σ ) || dσ | < + ∞ . α j = β j conditions then conditions ( i ) and ( ii ) are satisfiedif an only if U j is a half-plane of the form { z j ∈ C | Re ( z j e iαj ) > λ } for some λ ∈ R . If α j < β j , define ˜ x j = ˜ x j ( ζ j ) and ˜ y j = ˜ y j ( ζ j ) for ζ j ∈ C by the formula ζ j = z j + ˜ x j e ( − π − α j ) i + ˜ y j e ( π − β j ) i . (39)Notice that if ζ j ∈ U j , and if ˜ x j ( ζ ′ j ) ≥ ˜ x j ( ζ j ) and ˜ y j ( ζ ′ j ) ≥ ˜ y j ( ζ j ) then ζ ′ j ∈ z j + S ∗ α j ,β j ⊂ U j . This shows that there exists t j, ∈ [0 , s j, ] and t j, ∈ [0 , s j, ] and continuous piecewise C -functions f j and g j defined respectively on [0 , t j, ] and [0 , t j, ] such that U j − z j = { ζ j ∈ S ∗ α j ,β j | ˜ x ( ζ j ) ∈ (0 , t j, ] , ˜ y ( ζ j ) > f j (˜ x ( ζ j )) }∪{ ζ j ∈ S ∗ α j ,β j | ˜ x ( ζ j ) > t j, } = { ζ j ∈ S ∗ α j ,β j | ˜ y ( ζ j ) ∈ (0 , t j, ] , ˜ x ( ζ j ) > g j (˜ y ( ζ j )) } ∪ { ζ j ∈ S ∗ α j ,β j | ˜ y ( ζ j ) > t j, } . We have f j (0) = t j, , f j ( t j, ) = 0 , g j (0) = t j, , g j ( t j, ) = 0 , f j and g j arestrictly decreasing and f j = g − j if t j,s > for some, hence for all s ∈ { , } . For α = ( α , . . . , α k ) ∈ R k , β = ( β , . . . , β k ) ∈ R k , we will use the obviousconventions inf( α, β ) = (inf( α , β ) , . . . , inf( α k , β k )) , sup( α, β ) = (sup( α , β ) , . . . , sup( α k , β k )) . Clearly, (inf( α (1) , α (2) ) , sup( β (1) , β (2) ) ∈ M a,b if ( α (1) , β (1) ) ∈ M a,b and ( α (2) , β (2) ) ∈ M a,b . Proposition 12.2 If U (1) is admissible with respect to ( α (1) , β (1) ) ∈ M a,b andif U (2) is admissible with respect to ( α (2) , β (2) ) ∈ M a,b , then U (1) ∩ U (2) isadmissible with respect to (inf( α (1) , α (2) ) , sup( β (1) , β (2) )) . Set α (3) = inf( α (1) , α (2) ) , β (3) = sup( β (1) , β (2) ) , and set U (3) = U (1) ∩ U (2) . The fact that U (3) satisfies (i) follows from the fact that S ∗ α (3) j ,β (3) j = S ∗ α (1) j ,β (1) j ∩ S ∗ α (2) j ,β (2) j . The fact that U (3) satisfies (ii) follows easily from thefact that h z (1) + S ∗ α (1) ,β (1) i ∩ h z (2) + S ∗ α (2) ,β (2) i is itself admissible with respectto ( α (3) , β (3) ) if U (1) satisfies definition 12.1 with respect to z (1) and if U (2) satisfies definition 12.1 with respect to z (2) . (cid:3) Lemma 12.3
Let U be an admissible open set with respect to some ( α, β ) ∈ M a,b , and let F ∈ H (1) ( U ) . (i) We have, for ǫ = ( ǫ , . . . , ǫ k ) ∈ S ∗ α,β , Z Π j ≤ k ( U j \ ( U j + ǫ j ) ) | F ( ζ ) | dm ( ζ ) ≤ | ǫ | . . . | ǫ k |k F k H (1) ( U ) . where m denotes the Lebesgue measure on C k ≈ R k . (ii) We have, for ζ ∈ U, F ( ζ ) | ≤ k π k cos (cid:16) β − α (cid:17) . . . cos (cid:16) β k − α k (cid:17) dist ( ζ , ∂S ∗ α ,β ) . . . dist ( ζ k , ∂S ∗ α k ,β k )[ dist ( ζ , ∂U ) . . . dist ( ζ k , ∂U k )] k F k H (1) ( U ) . Proof: (i) Let F ∈ H (1) ( U ) , let ǫ = ( ǫ , . . . , ǫ k ) ∈ S ∗ α,β , for j ≤ k let γ j ∈ ( − π − α j , π − β j ) be a determination of arg ( ǫ j ) , and set r j = | ǫ j | > . Set U j, = z j + t j, e ( − π − α j ) i + S − π − α j ,γ j , U j, = z j + t j, e ( π − β j ) i + S γ j , π − β j , and U j, = z j + ∪ ρ> (cid:16) ρe γ j i + (cid:16) ∂U j ∩ S ∗ α j ,β j (cid:17)(cid:17) , with the convention U j, = ∅ if t j, = t j, = 0 . Also for ζ j ∈ C set x i = Re ( ζ j ) , y j = Im ( ζ j ) . For t j < , < ρ j < r j , set ζ j = ζ j ( ρ j , t j ) = ρ j e iγ j + ( t j, − t j ) e − i ( π − α j ) . This gives a parametrization of U j, \ ( U j, + ǫ j ) , and we have dx j dy j = (cid:12)(cid:12)(cid:12)(cid:12) cos ( γ j ) sin ( α j ) sin ( γ j ) cos ( α j ) (cid:12)(cid:12)(cid:12)(cid:12) dρ j dt j = cos ( α j + γ j ) dρ j dt j . Similarly for t j > t j, , < ρ j < r j , set ζ j = ζ j ( ρ j , t j ) = ρ j e iγ j + t j e i ( π − β j ) . This gives a parametrization of U j, \ ( U j, + ǫ j ) , and we have dx j dy j = (cid:12)(cid:12)(cid:12)(cid:12) cos ( γ j ) sin ( β j ) sin ( γ j ) cos ( β j ) (cid:12)(cid:12)(cid:12)(cid:12) dt j dρ j = cos ( β j + γ j ) dρ j dt j . Now assume that U j, = ∅ , so that t j, > and t j, > . For < t j < t j, , < ρ j < r j set ζ j = ζ j ( ρ j , t j ) = ρ j e iγ j + g j ( t j ) e ( − π − α j ) i + t j e ( π − β j ) i . This givesa parametrization of U j, \ ( U j, + ǫ j ) , and we have dx j dy j = (cid:12)(cid:12)(cid:12)(cid:12) cos ( γ j ) − g ′ j ( t ) sin ( α j ) + sin ( β j ) sin ( γ j ) − g ′ j ( t ) cos ( α j ) + cos ( β j ) (cid:12)(cid:12)(cid:12)(cid:12) dρ j dt j = ( cos ( β j + γ j ) − g ′ j ( t ) cos ( α j + γ j )) dρ j dt j . We have < cos ( α j + γ j ) < , < cos ( α j + γ j ) < , g ′ j ( t j ) < , and usingthe Cauchy-Schwartz inequality, we obtain < cos ( β j + γ j ) − g ′ j ( t ) cos ( α j + γ j )= cos ( γ j )( cos ( β j ) − g ′ j ( t ) cos ( α j )) − sin ( γ j )( sin ( β j ) − g ′ j ( t ) sin ( α j )) ≤ q ( cos ( β j ) − g ′ j ( t ) cos ( α j )) + ( sin ( β j ) − g ′ j ( t ) sin ( α j )) = q − g ′ j ( t ) cos ( β j − α j ) + g ′ j ( t ) . On the other hand we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ζ j ∂t j ( ρ j , t j ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:16) g ′ j ( t ) e ( − π − α j ) i + e ( π − β j ) i (cid:17) (cid:16) g ′ j ( t ) e ( π + α j ) i + e ( − π + β j ) i (cid:17) = 1 − g ′ j ( t ) cos ( β j − α j ) + g ′ j ( t ) . ∂U j + ρ j e iγ j being oriented from e ( − π − α j ) i . ∞ to e ( π − β j ) . ∞ , we obtain Z Π j ≤ k ( U j \ ( U j + ǫ j ) ) | F ( ζ ) | dm ( ζ ) ≤ Z (0 ,r ) × ... × (0 ,r k ) "Z Π j ≤ k ( ∂U j + ρ j e iγj ) | F ( σ j ) || dσ | . . . | dσ k | dρ . . . dρ k ≤ r . . . r k k F k H (1) ( U ) , which proves (i).(ii) Let F ∈ H (1) ( U ) , let ζ ∈ U , set r j = dist ( ζ j , ∂U j ) , set r = ( r , . . . , r k ) , and set B ( ζ, r ) = Π j ≤ k B ( ζ j , r j ) . Using Cauchy’s formula and polar coordinates,we obtain the standard formula F ( ζ ) = 1 | B ( ζ, r ) | Z B ( ζ,r ) F ( η ) dm ( η ) . (40)where | B ( ζ, r ) | = π k r . . . r k denotes the Lebesgue measure of B ( ζ, r ) . Denote by u j the orthogonal projection of ζ j on the real line z j + R e i ( − π − α j ) ,denote by v j the orthogonal projection of ζ j on the real line z j + R e ( π − β j ) i , andlet w j ∈ { u j , v j } be such that | ζ j − w j | = min( | ζ j − u j | , | ζ j − v j | ) . An easy topo-logical argument shows that w j ∈ ∂S ∗ α j ,β j , so that | ζ j − w j | = dist ( ζ j , ∂S ∗ α j ,β j ) ≥ dist ( ζ j , U j ) = r j . For λ ∈ R , we have ζ j / ∈ S ∗ α j ,β j + z j +2( ζ j − w j )+ λi ( ζ j − w j ) ⊃ U j + 2( ζ j − w j ) + λi ( ζ j − w j ) . If π − β j + α j > π , then ζ j − w j ∈ S ∗ α j ,β j . If π − β j + α j ≤ π , then we can choose λ ∈ R such that ζ j − w j + λi ( ζ j − w j ) ∈ S ∗ α j ,β j and such that | ζ j − w j + λi ( ζ j − w j ) | = | ζ j − w j | cos (cid:16) βj − αj (cid:17) . So there exists in both cases ǫ j ∈ S ∗ α j ,β j such that ζ j / ∈ U j + ǫ j and | ǫ j | = dist ( ζ j ,∂S ∗ αj,βj ) cos (cid:16) βj − αj (cid:17) . Using(40) and (i), we obtain | F ( ζ ) | ≤ π k r . . . r k Z Π j ≤ k ( U j \ ( U j + ǫ j ) ) | F ( η ) | dm ( η ) ≤ k π k cos (cid:16) β − α (cid:17) . . . cos (cid:16) β k − α k (cid:17) dist ( ζ , ∂S ∗ α ,β ) . . . dist ( ζ k , ∂S ∗ α k ,β k )[ dist ( ζ , ∂U ) . . . dist ( ζ k , ∂U k )] k F k H (1) ( U ) , which proves (ii). (cid:3) Corollary 12.4 ( H (1) ( U ) , k . k H (1) ( U ) ) is a Banach space, F U + ǫ is bounded on U + ǫ, and lim dist ( ζ,∂U ) → + ∞ ζ ∈ U + ǫ F ( ζ ) = 0 for every F ∈ H (1) ( U ) and every ǫ ∈ S ∗ α,β . Proof: It follows from (ii) that for every compact set K ⊂ U there exists m K > such that max ζ ∈ K | F ( ζ ) | ≤ m K k F k H (1) ( U ) . So every Cauchy sequence73 F n ) n ≥ in ( H (1) ( U ) , k . k H (1) ( U ) ) is a normal family which converges uniformlyon every compact subset of U to a holomorphic function F : U → C . Since R ˜ ∂U + ǫ | F ( σ ) − F n ( σ ) || dσ | = lim R → + ∞ R B (0 ,R ) ∩ ( ˜ ∂U + ǫ ) | F ( σ ) − F n ( σ ) | dσ | , an easyargument shows that F ∈ H (1) ( U ) and that lim n → + ∞ k F − F n k H (1) ( U ) = 0 . Now let ǫ > . There exists m j > such that dist ( ζ j , ∂U ) ≥ m j dist ( ζ j , ∂S ∗ α j ,β j ) for every ζ j ∈ U j + ǫ j , which gives, for ζ ∈ U + ǫ, | F ( ζ ) | ≤ k π k m . . . m k cos (cid:16) β − α (cid:17) . . . cos (cid:16) β k − α k (cid:17) dist ( ζ , ∂U ) . . . dist ( ζ k , ∂U k ) k F k H (1) ( U ) . Since inf ζ j ∈ U j + ǫ j dist ( ζ j , ∂U j ) > for j ≤ k, this shows that F is boundedon U + ǫ, and that lim dist ( ζ,∂U ) → + ∞ ζ ∈ U + ǫ F ( ζ ) = 0 . (cid:3) Theorem 12.5
Let U be an admissible open set with respect to some ( α, β ) ∈ M a,b , and let F ∈ H (1) ( U ) . Then Z ˜ ∂U + ǫ F ( σ ) dσ = 0 for every ǫ ∈ S ∗ α,β , and F ( ζ ) = 1(2 iπ ) k Z ˜ ∂U + ǫ F ( σ ) dσ ( ζ − σ ) . . . ( ζ k − σ k ) for every ǫ ∈ S ∗ α,β and for every ζ ∈ U + ǫ, where ∂U j is oriented from e i ( − π − α j ) . ∞ to e i ( π − βj ) . ∞ for j ≤ k. Proof: Let z ∈ C k satisfying the conditions of definition 12.1 with respectto U, let ǫ ∈ S ∗ α,β , let L > such that ( z j + e iα j . ∞ , z j + Le iα j ] ⊂ ∂U j and [ z j + Le iβ j , z j + e iβ j . ∞ ) ⊂ ∂U j for j ≤ k, and let M > . Set Γ L,j, = ( ǫ j + ∂U j ) \ (cid:0) ( z j + ǫ j + Le iα j , z j + ǫ j + e iα j . ∞ ) ∪ ( z j + ǫ j + Le iβ j , z j + ǫ j + e iβ j . ∞ ) (cid:1) , Γ L,j, = [ z j + ǫ j + Le iβ j , z j + Lǫ j + Le iβj ] , Γ L,j, = ( Lǫ j + ∂U j ) \ (cid:0) ( z j + Lǫ j + Le iα j . ∞ , z j + Lǫ j + Le iα j ) ∪ ( z j + Lǫ j + Le iβ , z j + ǫ j + e iβ j . ∞ ) (cid:1) , Γ L,j, = [ z j + Lǫ j + Le iα j , z j + ǫ j + Le iα j ] , Γ L,j = ∪ ≤ s ≤ Γ L,j,s , where the Jordan curve Γ L,j is oriented clockwise.For n ≥ , ζ j ∈ S ∗ α j ,β j , set W j,n ( ζ j ) = n n + e αj + βj i ζ j ! , and set W n ( ζ ) =Π j ≤ k W n,j ( ζ j ) for ζ ∈ S ∗ α,β . Then | W n,j ( ζ j ) | ≤ for ζ j ∈ S ∗ α j ,β j , W n ( ζ ) → as n → ∞ uniformly on compact sets of S ∗ α,β , and lim | ζ |→∞ ζ ∈ S ∗ α,β W n ( ζ ) = 0 . Denote by V L,j the interior of Γ L,j and set V L = Π ≤ j ≤ k V L,j . If ζ ∈ V L , itfollows from Cauchy’s theorem that we have X l ∈{ , , , } k Z Π j ≤ k Γ L,j,l ( j ) W n ( σ − z − ǫ ) F ( σ ) dσ = Z ˜ ∂V L W n ( σ − z − ǫ ) F ( σ ) dσ = 0 . l ( j ) = 1 for j ≤ k. It follows from the corollary that there exists
M > such that | F ( ζ ) | ≤ M for ζ ∈ U + ǫ, and there exists R n > such that R Γ L,j | W n ( σ j − z j − ǫ j ) || dσ | ≤ R n for every L. Also lim sup L → + ∞ R Γ L,j,s | W n,j ( σ j − z j − ǫ j ) || d ( σ j ) | = 0 for s ≥ , j ≤ k. Let l = l , and let j l ≤ k such that j l ≥ . We have lim sup L → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Π j ≤ k Γ L,j,l ( j ) | W n ( σ − z − ǫ ) F ( σ ) dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M R k − n Z Γ L,jl,l ( jl ) | W n,j l ( ζ j l − z j l − ǫ j l ) || dσ j l | = 0 . This gives Z ∂U + ǫ W n ( σ − z − ǫ ) F ( σ ) dσ = lim L → + ∞ Z Π j ≤ k Γ L,j,l j ) W n ( σ − z − ǫ ) F ( σ ) dσ = lim L → + ∞ X l ∈{ , , , } k Z Π j ≤ k Γ L,j,l ( j ) W n ( σ − z − ǫ ) F ( σ ) dσ = 0 . It follows then from the Lebesgue dominated convergence theorem that R ∂U + ǫ F ( σ ) dσ = 0 . Similarly, applying Cauchy’s formula when ζ ∈ U + ǫ is contained in V L , weobtain iπ ) k Z ∂U + ǫ W n ( σ − z − ǫ ) F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ = lim L → + ∞ iπ ) k Z Γ L,j,l j ) W n ( σ − z − ǫ ) F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ = lim L → + ∞ iπ ) k X l ∈{ , , , } k Z Π j ≤ k Γ L,j,l ( j ) W n ( σ − z − ǫ ) F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ = lim L → + ∞ iπ ) k Z ˜ ∂V L W n ( σ − z − ǫ ) F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ = W n ( ζ − z − ǫ ) F ( ζ ) . It follows then again from the Lebesgue dominated convergence theorem thatwe have F ( ζ ) = lim n → + ∞ W n ( ζ − z − ǫ ) F ( ζ ) = lim n → + ∞ iπ ) k Z ∂U + ǫ W n ( σ − z − ǫ ) F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ = 1(2 iπ ) k Z ∂U + ǫ F ( σ )( ζ − σ ) . . . ( ζ k − σ k ) dσ. (cid:3) Let ζ ∈ U, and let ǫ ∈ S ∗ α,β . It follows from the theorem that there ex-ists ρ > such that | F ( ζ ) | ≤ π ) k k F k H (1)( U ) Π j ≤ k dist ( ζ j ,∂U j + tǫ j ) for t ∈ (0 , ρ ] . Since75 im t → + dist ( ζ j , ∂U j + tǫ j ) = lim t → + dist ( ζ j − tǫ j , ∂U j ) = dist ( ζ j , ∂U j ) , weobtain, for F ∈ H (1) ( U ) , ζ ∈ U, | F ( ζ ) | ≤ π ) k k F k H (1) ( U ) Π j ≤ k dist ( ζ j , ∂U j ) (41)which improves inequality (ii) of lemma 12.3.If α j = β j for j ≤ k , then every ( α, β ) admissible open set U is a product ofopen half-planes and the space H (1) ( U ) is the usual Hardy space H ( U ) . Thestandard conformal mappings of the open unit disc D onto half planes inducean isometry from the Hardy space H ( D k ) onto H ( U ) . It follows then fromstandard results about H ( D k ) , see theorems 3.3.3 and 3.3.4 of [28], that F admits a.e. a nontangential limit F ∗ on ˜ ∂U, and that lim ǫ → | R ˜ ∂U | F ∗ ( σ ) − F ( σ + ǫ ) || dσ | = 0 . This gives the formula F ( ζ ) = 1(2 iπ ) k Z ˜ ∂U F ∗ ( σ ) dσ ( ζ − σ ) . . . ( ζ k − σ k ) for every ζ ∈ U. (42)We did not investigate whether such nontangential limits of F on ˜ ∂U existin the general case.Recall that the Smirnov class N + ( P + ) on the right-hand open half-plane P + consists in those functions F holomorphic on P + which can be written underthe form F = G/H where G ∈ H ∞ ( P + ) and where H ∈ H ∞ ( P + ) is outer,which means that we have, for Re ( ζ ) > ,F ( ζ ) = exp (cid:18) π Z + ∞−∞ − iyζ ( ζ − iy )(1 + y ) log | F ∗ ( iy ) | dy (cid:19) , where F ∗ ( iy ) = lim x → + F ( x + iy ) if defined a.e. on the vertical axis andsatisfies R + ∞−∞ | log | F ∗ ( iy ) | y dy < + ∞ . Set , for Re ( ζ ) > ,F n ( ζ ) = exp (cid:18) π Z + ∞−∞ − iyζ ( ζ − iy )(1 + y ) sup( log | F ∗ ( iy ) | , − n ) dy (cid:19) . It follows from the positivity of the Poisson kernel on the real line that | F ( ζ ) | ≤ | F n ( ζ ) | and that lim n → + ∞ F n ζ ) = F ( ζ ) for Re ( ζ ) > . Also the non-tangential limit F ∗ n ( iy ) of F at iy exists a.e. on the imaginary axis and | F ∗ n ( iy ) | =sup( e − n , | F ∗ ( iy ) | ) a.e., which shows that sup ζ ∈ P + | F n ( ζ ) | = sup ζ ∈ P + | F ( ζ ) | when n is sufficiently large. Hence lim n → + ∞ F ( ζ ) F − n ( ζ ) = 1 for ζ ∈ P + . This suggests the following notion;
Definition 12.6
Let U ⊂ C k be a connected open set. A holomorphic function F ∈ H ∞ ( U ) is said to be strongly outer on U if there exists a sequence ( F n ) n ≥ of invertible elements of H ∞ ( U ) satisfying the following conditions(i) | F ( ζ ) | ≤ | F n ( ζ ) | ( ζ ∈ U, n ≥ , (ii) lim n → + ∞ F ( ζ ) F − n ( ζ ) = 1 ( ζ ∈ U ) . he Smirnov class S ( U ) consists of those holomorphic functions F on U such that F G ∈ H ∞ ( U ) for some strongly outer function G ∈ H ∞ ( U ) . It follows from (ii) that F ( ζ ) = 0 for every ζ ∈ U if F is strongly outer on U, and F | V is strongly outer on V if V ⊂ U. Similarly if F ∈ S ( U ) then F | V ∈ S ( V ) . Also it follows immediately from the definition that the set of bounded stronglyouter functions on U is stable under products, and that if there is a conformalmapping θ from an open set V ⊂ C k onto U then F ∈ H ∞ ( U ) is strongly outeron U if and only if F ◦ θ is strongly outer on V , and if G is holomorphic on U then G ∈ S ( U ) if and only F ◦ θ ∈ S ( V ) . Now let ( α, β ) ∈ M a,b and let U = Π j ≤ k U j be an admissible open set withrespect to ( α, β ) . Then each set U j is conformally equivalent to the open unitdisc D , and so there exists a conformal mapping θ from D k onto U , and thestudy of the class of bounded strongly outer functions on U (resp. the Smirnovclass on U ) reduces to the study of the class bounded strongly outer functions(resp. the Smirnov class) on D k . Let F ∈ H ∞ ( D k ) be strongly outer, and let ( F n ) n ≥ be a sequence of invert-ible elements of H ∞ ( D k ) satisfying the conditions of definition 12.6 with respectto F. Denote by T = ∂D the unit circle. Then H ∞ ( D k ) can be identified to a w ∗ -closed subspace of L ∞ ( T k ) with respect to the w ∗ -topology σ ( L ( T k ) , L ∞ ( T k )) . Let L ∈ H ∞ ( D k ) be a w ∗ -cluster point of the sequence ( F F − n ) n ≥ . Sincethe map G → G ( ζ ) is w ∗ -continuous on H ∞ ( D k ) for ζ ∈ D k , L = 1 , andso F H ∞ ( D k ) is w ∗ -dense in H ∞ . When k = 1 , this implies as well-known that F is outer, and the argument used for the half-plane shows that, conversely,every bounded outer function on D is strongly outer, and so S ( D ) = N + ( D ) . Recall that a function G ∈ H ∞ ( D k ) is said to be outer if log ( | G (0 , . . . , | ) = π ) k R T k log | G ∗ ( e it , . . . , e it k ) | dt . . . dt k , where G ∗ ( e it , . . . , e it k ) denotes a.e. thenontangential limit of G at ( e it , . . . , e it k ) , see [28], definition 4.4.3, and G is outer if and only if almost every slice function G ω is outer on D , where G ω ( ζ ) = G ( ωζ ) for ω ∈ T k , ζ ∈ D , see [28], lemma 4.4.4. If follows from defi-nition 12.6 that every slice function F ω is strongly outer on D if F ∈ H ∞ ( D k ) is strongly outer on D k , and so every strongly outer bounded function on D k isouter. It follows from an example from [28] that the converse is false if k ≥ . Proposition 12.7
Let k ≥ , and set F ( ζ , . . . , ζ k ) = e ζ ζ ζ ζ − for ( ζ , . . . , ζ k ) ∈ D k . Then F is outer on D k , but F is not strongly outer on D k . Proof: Set f ( ζ ) = e ζ +1 ζ − for ζ ∈ D . Then f ∈ H ∞ ( D ) is a singular innerfunction. Since f ( ζ ) = 0 for ζ ∈ D , it follows from [28], lemma 4.4.4b that thefunction ˜ f : ( ζ , ζ ) → f ( ζ + ζ , ζ + ζ ) = e ζ ζ ζ ζ − is outer on D . Hence we have log | F (0 , . . . , | = log | ˜ f (0 , | = 1(2 π ) Z T ˜ f ( e it , e it ) dt dt = 1(2 π ) k Z T k F ( e it , . . . , e it k ) dt . . . dt k , F is outer on D k . Now set ω = (1 , . . . , . Then F ω = f is not outer on D , and so F is notstrongly outer on D k . (cid:3) The fact that some bounded outer functions on D are not strongly outer isnot surprising: The Poisson intergal of a real valued integrable function on T k is the real part of some holomorphic function on D k if an only if its Fouriercoefficients vanish on Z k \ ( Z + ) k ∪ ( Z − ) k , see [28], theorem 2.4.1, and so theconstruction of the sequence ( F n ) n ≥ satisfying the conditions of definition 12.6with respect to a bounded outer function F on D k breaks down when k ≥ . We conclude this appendix with the following trivial observations.
Proposition 12.8
Let U = Π j ≤ k U j ⊂ C k be an admissible open set with respectto some ( α, β ) ∈ M a,b . (i) Let θ j : U j → D be a conformal map and let π j : ( ζ , . . . , ζ k ) → ζ j be thej-th cooordinate projection. If f ∈ H ∞ ( D ) is outer, then f ◦ θ j ◦ π j is stronglyouter on U. (ii) The Smirnov class S ( U ) contains all holomorphic functions on U havingpolynomial growth at infinity. Proof: (i) Since f is strongly outer on D, there exists a sequence ( f n ) n ≥ ofinvertible elements of H ∞ ( D ) satisfying the conditions of definition 12.6 withrespect to f. Then the sequence ( f n ◦ θ j ◦ π j ) n ≥ satisfies the conditions ofdefinition 12.6 with respect to f ◦ θ j ◦ π j , and so f ◦ θ j ◦ π j is strongly outer on U. (ii) For j ≤ k there exists γ j ∈ [ − π, π ) and m j ∈ R such that open set U j is contained in the open half plane P j := { ζ j ∈ C | Re ( ζ j e iγ j ) ≥ m j } . Thefunction σ → − σ is outer on D , since | − σ | ≤ (cid:12)(cid:12)(cid:12) /n − σ (cid:12)(cid:12)(cid:12) for σ ∈ D , and thefunction ζ j → ζ j e iγj − m j − ζ j e iγj − m j +1 maps conformally U j onto D . Set F j ( ζ , . . . , ζ k ) = − ζjeiγj − mj − ζjeiγj − mj +1 = ζ j e iγj − m j +1 . It follows from (i) that F j is strongly outer on Π j ≤ k P j , hence strongly outer on U. Now assume that a function F holomorphic on U has polynomial growth atinfinity. Then there exists p ≥ such that F Π j ≤ k F pj is bounded on U, and so F ∈ S ( U ) . (cid:3) References [1] G.R. Allan, H.G. Dales and P. Mc Clure,
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