Holomorphic Functions and polynomial ideals on Banach spaces
aa r X i v : . [ m a t h . F A ] F e b HOLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES
DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO
Abstract.
Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra ofholomorphic functions of bounded type, H b A ( E ) . We prove that, under very natural conditions satisfied bymany usual classes of polynomials, the spectrum M b A ( E ) of this algebra “behaves” like the classical case of M b ( E ) (the spectrum of H b ( E ) , the algebra of bounded type holomorphic functions). More precisely, weprove that M b A ( E ) can be endowed with a structure of Riemann domain over E ′′ and that the extensionof each f ∈ H b A ( E ) to the spectrum is an A -holomorphic function of bounded type in each connectedcomponent. We also prove a Banach-Stone type theorem for these algebras. Introduction
We consider algebras of analytic functions associated to sequences of polynomial ideals. More precisely,if A = { A k } k is a coherent sequence of Banach polynomial ideals (see the definitions below) and E isa Banach space, we consider the space H b A ( E ) of holomorphic functions of bounded type associated to A ( E ) , much in the spirit of holomorphy types introduced by Nachbin [33] (see also [23]). These spacesof A -holomorphic functions of bounded type were introduced in [11], and many particular classes ofholomorphic functions appearing in the literature are obtained by this procedure from usual sequences ofpolynomial ideals (such as nuclear, integral, approximable, weakly continuous on bounded sets, extendible,etc.). Under certain multiplicativity conditions on the sequence of polynomial ideals (satisfied for all thementioned examples), H b A ( E ) turns out to be an algebra.We study this multiplicativity condition and relate it with properties of the associated tensor norms.We show that polynomial ideals associated to natural symmetric tensor norms (in the sense of [14]) aremultiplicative. We also prove that composition and maximal/minimal hulls of multiplicative sequences ofpolynomial ideals are multiplicative.Whenever H b A ( E ) is an algebra, we study its spectrum M b A ( E ) and show, under fairly general as-sumptions, that it has an analytic structure as a Riemann domain over E ′′ , a result analogous to thatfor the spectrum M b ( E ) of H b ( E ) , the algebra of all holomorphic functions of bounded type (see [4] and[22, Section 6.3]). Moreover, the connected components of M b A ( E ) are analytic copies of E ′′ , and onemay wonder if the Gelfand extension of a function f to M b A ( E ) is analytic and, also, if the restriction ofthis extension to each connected component can be thought as a function in H b A ( E ′′ ) . To answer thesequestions, we study the Aron-Berner extension of functions in H b A ( E ) and also translation and convo-lution operators on these kinds of algebras. We obtain conditions on the sequence of polynomial idealsthat ensure a positive answer to both questions, which are satisfied by most of the examples consideredthroughout the article.Finally, we address a Banach-Stone type question on these algebras: if H b A ( E ) and H b B ( F ) are (topo-logically and algebraically) isomorphic, what can we say about E and F ? We obtain results in this Mathematics Subject Classification.
Key words and phrases.
Polynomial ideals, holomorphic functions, Riemann domains over Banach spaces.Partially supported by ANPCyT PICT 05 17-33042. The first and third authors were also partially supported by UBACyTGrant X038 and ANPCyT PICT 06 00587. direction which allow us to show, for example, that if E or F is reflexive and A and B are any of thesequence of nuclear, integral, approximable or extendible polynomials, then if H b A ( E ) is isomorphic to H b B ( F ) it follows that E and F are isomorphic.We refer to [22, 32] for notation and results regarding polynomials and holomorphic functions in general,to [25, 26] for polynomial ideals and to [20, 24] for tensor products of Banach spaces.1. Preliminaries
Throughout this paper E will denote a complex Banach space and P k ( E ) is the Banach space of allcontinuous k -homogeneous polynomials from E to C . If P ∈ P k ( E ) , there exists a unique symmetric n -linear mapping ∨ P : E × · · · × E | {z } k → C such that P ( x ) = ∨ P ( x, . . . , x ) . We define, for each a ∈ E , P a j ∈ P k − j ( E ) by P a j ( x ) = ∨ P ( a j , x k − j ) = ∨ P ( a, ..., a | {z } j , x, ..., x | {z } k − j ) . For j = 1 , we write P a instead of P a .Let us recall the definition of polynomial ideals [25, 26]. A Banach ideal of (scalar-valued) con-tinuous k -homogeneous polynomials is a pair ( A k , k · k A k ) such that:(i) For every Banach space E , A k ( E ) = A k ∩ P k ( E ) is a linear subspace of P k ( E ) and k · k A k ( E ) is anorm on it. Moreover, ( A k ( E ) , k · k A k ( E ) ) is a Banach space.(ii) If T ∈ L ( E , E ) and P ∈ A k ( E ) , then P ◦ T ∈ A k ( E ) with k P ◦ T k A k ( E ) ≤ k P k A k ( E ) k T k k . (iii) z z k belongs to A k ( C ) and has norm 1.In [12] we defined and studied coherent sequences of polynomial ideals. Here we present more resultsabout this topic. Even though the original definitions were for general vector valued polynomial ideals,we focus in this article on the case of scalar valued polynomial ideals.We recall the definitions: Definition 1.1.
Consider the sequence A = { A k } ∞ k =1 , where for each k , A k is a Banach ideal of scalarvalued k -homogeneous polynomials. We say that { A k } k is a coherent sequence of polynomial ideals if there exist positive constants C and D such that for every Banach space E , the following conditionshold for every k ∈ N : (i) For each P ∈ A k +1 ( E ) and a ∈ E , P a belongs to A k ( E ) and k P a k A k ( E ) ≤ C k P k A k +1 ( E ) k a k (ii) For each P ∈ A k ( E ) and γ ∈ E ′ , γP belongs to A k +1 ( E ) and k γP k A k +1 ( E ) ≤ D k γ kk P k A k ( E ) OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 3
In [12], many examples of coherent sequences were presented (see Examples below). Let us see nowthat from any pair of such examples, it is “easy” to construct many others.If A k and A k are Banach ideals of k -homogeneous polynomials, for each < θ < , we denote by A θk the polynomial ideal defined by A θk ( E ) = (cid:2) A k ( E ) , A k ( E ) (cid:3) θ for every Banach space E. That is, (cid:2) A k ( E ) , A k ( E ) (cid:3) θ is the space obtained by complex interpolation from the pair (cid:0) A k ( E ) , A k ( E ) (cid:1) with parameter θ . The complex interpolation can be defined because both spaces are included in thespace of continuous k -homogeneous polynomials P k ( E ) . Note also that A θk is actually a Banach ideal bythe properties of interpolations spaces, since the ideal properties can be rephrased as the continuity ofcertain linear operators.The construction of interpolating spaces easily implies the following result. Proposition 1.2.
Let { A k } k and { A k } k be coherent sequences of polynomial ideals with constants C , D and C , D , respectively. Then, for every < θ < , the sequence { A θk } k is coherent with constants C − θ C θ and D − θ D θ . There is a natural way of building a class of holomorphic functions associated to a coherent sequenceof polynomial ideals. In [11] we defined:
Definition 1.3.
Let A = { A k } k be a coherent sequence of polynomial ideals and E be a Banach space.We define the space of A -holomorphic functions of bounded type by H b A ( E ) = (cid:26) f ∈ H ( E ) : d k f (0) k ! ∈ A k ( E ) and lim k →∞ (cid:13)(cid:13)(cid:13) d k f (0) k ! (cid:13)(cid:13)(cid:13) /k A k ( E ) = 0 (cid:27) . We define in H b A ( E ) the seminorms p R , for R > , by p R ( f ) = ∞ X k =0 (cid:13)(cid:13)(cid:13) d k f (0) k ! (cid:13)(cid:13)(cid:13) A k ( E ) R k , for f ∈ H b A ( E ) . It is proved in [11] that this is a Fréchet space. Since for every polynomial P ∈ A k ( E ) we have k P k ≤ k P k A k ( E ) , the space H b A ( E ) is continuously contained in H b ( E ) .The following examples of spaces of holomorphic functions of bounded type were already defined in theliterature and can be seen as particular cases of the above definition. Example 1.4. (a) Let A be the sequence of continuous homogeneous polynomials A k = P k , k ≥ .Then H b A ( E ) = H b ( E ) .(b) If A is the sequence of weakly continuous on bounded sets polynomial ideals then H b A ( E ) is thespace of weakly uniformly continuous holomorphic functions of bounded type H bw ( E ) defined byAron in [2].(c) If A is the sequence of nuclear polynomial ideals then H b A ( E ) is the space of nuclearly entirefunctions of bounded type H Nb ( E ) defined by Gupta and Nachbin (see [22, 28]).(d) If A is the sequence of extendible polynomials, A k = P ke , k ≥ . Then, by [9, Proposition 14], H b A ( E ) is the space of all f ∈ H ( E ) such that, for any Banach space G ⊃ E , there is an extension ˜ f ∈ H b ( G ) of f .(e) Let A be the sequence of integral polynomials, A k = P kI , k ≥ . Then H b A ( E ) is the space ofintegral holomorphic functions of bounded type H bI ( E ) defined in [21]. DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO Multiplicative sequences
Definition 2.1.
Let { A k } k be a sequence of scalar valued polynomial ideals. We will say that { A k } k is multiplicative if it is coherent and there exists a constant M such that for each P ∈ A k ( E ) and Q ∈ A l ( E ) , we have that P Q ∈ A k + l ( E ) and k P Q k A k + l ( E ) ≤ M k + l k P k A k ( E ) k Q k A l ( E ) . Our interest in multiplicative sequences of polynomial ideals is motivated by the following result from[11]:
Lemma 2.2.
Let A be a multiplicative sequence. If f, g ∈ H b A ( E ) then f · g ∈ H b A ( E ) . Therefore, H b A ( E ) is an algebra. In the following section we study the spectrum of this algebra. Now, let us see some examples ofmultiplicative sequences.
Example 2.3. (a) If A k is the ideal of all k -homogeneous (or of approximable, extendible, weaklycontinuous on bounded sets) polynomials then { A k } k is a multiplicative sequence.(b) If A k is the ideal of all k -homogeneous nuclear polynomials then { A k } k is a multiplicative sequence(see for example [22, Exercise 2.63] or deduced it as a consequence of the following example andCorollary 2.8).(c) If A k is the ideal of all k -homogeneous integral polynomials then { A k } k is a multiplicative sequence.Indeed, for P ∈ P I ( k E ) , Q ∈ P I ( l E ) , let us prove that P Q is a continuous linear functionalon the k -fold symmetric tensor product of E with the injective symmetric norm ( ε sk ). Take ψ = P i x k + li ∈ N k + l,sε sk + l E , then (cid:12)(cid:12) h P Q, ψ i (cid:12)(cid:12) = (cid:12)(cid:12) X i P ( x i ) Q ( x i ) (cid:12)(cid:12) = (cid:12)(cid:12) P (cid:0) X i x ki Q ( x i ) (cid:1)(cid:12)(cid:12) ≤ k P k I sup γ ∈ B E ′ (cid:12)(cid:12) X i γ ( x i ) k Q ( x i ) (cid:12)(cid:12) = k P k I sup γ ∈ B E ′ (cid:12)(cid:12)(cid:12) Q (cid:16) X i γ ( x i ) k x li (cid:17)(cid:12)(cid:12)(cid:12) ≤ k P k I k Q k I sup γ ∈ B E ′ sup ϕ ∈ B E ′ (cid:12)(cid:12) X i γ ( x i ) k ϕ ( x i ) l (cid:12)(cid:12) . Let R ∈ P ( k + l E ′ ) , R ( γ ) := P γ ( x i ) k + l for γ ∈ E ′ . Then by [29, Corollary 4], if we take γ, ϕ ∈ S E ′ ,we obtain | ∨ R ( γ, . . . , γ | {z } k , ϕ, . . . , ϕ | {z } l ) | = (cid:12)(cid:12) X i γ ( x i ) k ϕ ( x i ) l (cid:12)(cid:12) ≤ ( k + l ) k + l ( k + l )! k ! k k l ! l l k R k = ( k + l ) k + l ( k + l )! k ! k k l ! l l ε sk + l (cid:16) X i x k + li (cid:17) . Therefore, (cid:12)(cid:12) h P Q, ψ i (cid:12)(cid:12) ≤ ( k + l ) k + l ( k + l )! k ! k k l ! l l k P k I k Q k I ε sk + l ( ψ ) , and so P Q is integral with k P Q k I ≤ ( k + l ) k + l ( k + l )! k ! k k l ! l l k P k I k Q k I ≤ e k + l k P k I k Q k I .S. Lassalle and C. Boyd proved that the product of Banach algebra valued integral polynomialsis integral (personal communication).One may wonder if any coherent sequence is automatically multiplicative. The answer is no. Theconstruction in [12, Section 2] can be easily adapted to obtain a coherent sequence { A n } n with A = L , A = P , A = P and A = P wsc (the -homogeneous polynomials that are weakly sequentially OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 5 continuous at 0). To see that the sequence is not multiplicative consider, for instance, P ∈ P ( ℓ ) givenby P ( x ) = P n x n . Then P ∈ A ( ℓ ) but P A ( ℓ ) .Suppose we have a sequence of ideals which is related to a sequence of tensor norms. In this case, themultiplication property of the ideal has a translation into properties of the tensor norms, as shown inProposition 2.6 below. First we need the following proposition, that we believe is of independent interest.Recall that a normed ideal of linear operators is said to be closed if the norm considered is the usualoperator norm. Proposition 2.4.
Let { A k } k be a sequence of polynomial ideals and C be a closed ideal of operators.Suppose that there exists a constant M such that for each P ∈ A k ( E ) and Q ∈ A l ( E ) it holds that P Q ∈ A k + l ( E ) with k P Q k A k + l ( E ) ≤ M k P k A k ( E ) k Q k A l ( E ) . Then, the sequence { A k ◦ C } k has the same property.Proof. Take P ∈ A k ◦ C ( E ) and Q ∈ A l ◦ C ( E ) and write them as P = ˜ P ◦ S and Q = ˜ Q ◦ T , with S ∈ C ( E, E ) , T ∈ C ( E, E ) , k S k C ( E,E ) = k T k C ( E,E ) = 1 , ˜ P ∈ A k ( E ) and ˜ Q ∈ A l ( E ) . We considerthe product space E × E with the supremum norm and define ˜ S : E → E × E and ˜ T : E → E × E by ˜ S ( x ) = ( S ( x ) , and ˜ T ( x ) = (0 , T ( x )) . Clearly, ˜ S and ˜ T belong to C ( E, E × E ) and so is ˜ S + ˜ T .Moreover, the norm of ˜ S + ˜ T in C is the maximum of those of S and T , thus k ˜ S + ˜ T k C ( E,E × E ) = 1 .On the other hand, in a similar way we can see that R : E × E → K given by R ( y , y ) = ˜ P ( y ) ˜ Q ( y ) belongs to A k + l ( E × E ) , and k R k A k + l ( E × E ) ≤ M k ˜ P k A k ( E ) k ˜ Q k A l ( E ) . Since P Q = R ◦ ( ˜ S + ˜ T ) , wehave that P Q belongs to A k + l ◦ C . Moreover, k P Q k A k + l ◦ C ( E ) ≤ k R k A k + l ( E × E ) k ˜ S + ˜ T k C ( E,E × E ) ≤ M k ˜ P k A k ( E ) k ˜ Q k A l ( E ) . Considering all the possible factorizations of P and Q (with operators of norm 1) we obtain the desirednorm estimate. (cid:3) Corollary 2.5.
Let { A k } k be a multiplicative sequence and C a closed ideal of operators. Then { A k ◦ C } k is a multiplicative sequence.Proof. Just combine Proposition 2.4 and [12, Proposition 3.1]. (cid:3)
Now we turn our attention to tensor norms. We say that a polynomial ideal A k is associated to thefinitely generated k -fold symmetric tensor norm α k if for each finite dimensional normed space M we havethe isometry A k ( M ) : = (cid:0) ⊗ k,s M ′ , α k (cid:1) . It is clear that each tensor norm has unique maximal and minimal associated ideals.The following result is the announced translation of the multiplication property into a tensorial setting.By σ we denote the symmetrization operator on the corresponding tensor product. Proposition 2.6.
For each k natural numbers, let α k be finitely generated k -fold symmetric tensor norm α k . Consider A maxk and A mink the maximal and minimal ideals associated to α k . Fixed c k,l > , thefollowing assertions are equivalent.( i ) For every Banach space E , if P ∈ A maxk ( E ) and Q ∈ A maxl ( E ) then P Q ∈ A maxk + l ( E ) and k P Q k A maxk + l ( E ) ≤ c k,l k P k A maxk ( E ) k Q k A maxl ( E ) . DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO ( ii ) For every Banach space E , if P ∈ A mink ( E ) and Q ∈ A minl ( E ) then P Q ∈ A mink + l ( E ) and k P Q k A mink + l ( E ) ≤ c k,l k P k A mink ( E ) k Q k A minl ( E ) . ( iii ) For every Banach space E , if s ∈ N k,sα k E ′ and t ∈ N l,sα l E ′ , then α k + l ( σ ( s ⊗ t )) ≤ c k,l α k ( s ) α l ( t ) . Proof.
The three statements are clearly equivalent if E is a finite dimensional Banach space. By the verydefinition of maximal polynomial ideals [27], if ( i ) holds for finite dimensional Banach spaces, then it alsoholds for every Banach space. As a consequence, ( i ) is implied by either ( ii ) or by ( iii ).We now prove that ( i ) implies ( iii ). Note that ( iii ) is equivalent to prove that the bilinear map φ E : (cid:16) N k,sα k E ′ × N l,sα l E ′ , k · k ∞ (cid:17) → N k + l,sα k + l E , φ E ( s, t ) = σ ( s ⊗ t ) is continuous of norm ≤ c k,l for everyBanach space E . If ( i ) is true then φ S is continuous (with norm ≤ c k,l ) for every finite dimensional Banachspace S . Let M, N be two finite dimensional subspaces of E ′ such that s ∈ N k,s M and t ∈ N l,s N . Then α k + l (cid:0) σ ( s ⊗ t ) , O k + l,s M + N (cid:1) ≤ c k,l max { α k (cid:0) s, O k,s M + N (cid:1) , α l (cid:0) t, O l,s M + N (cid:1) }≤ c k,l max { α k (cid:0) s, O k,s M (cid:1) , α l (cid:0) t, O l,s N (cid:1) } , where the second inequality is true by the metric mapping property. Taking the infimum over M and N we obtain that k φ E k ≤ c k,l and thus we have ( iii ).To see that ( i ) implies ( ii ) , just note that A mink = A maxk ◦ F and use Proposition 2.4. (cid:3) Remark 2.7. (a) Note that in the proof of the previous proposition, we have shown that if { A k } k or { α k } k satisfy any of the three conditions on spaces of finite dimension, then the three statements ofprevious proposition hold for every Banach space.(b) The analogous statement holds for the coherence conditions in Definition 1.1.(c) Condition (iii) is one of the inequalities fulfilled by a “family of complemented symmetric semi-norms”, defined by C. Boyd and S. Lassalle in [6] . Corollary 2.8.
Let { A k } k be a multiplicative sequence. Then { A mink } k and { A maxk } k are multiplicativesequences. In [14], natural tensor norms for arbitrary order are introduced and studied, in the spirit of the naturaltensor norms of Grothendieck. Let us see that the polynomial ideals associated to the natural symmetrictensor norms are multiplicative. First we introduce some notation.For a symmetric tensor norm β k (of order k ), the projective and injective associates (or hulls) of β k will be denoted, by extrapolation of the 2-fold case, as \ β k / and /β k \ respectively. They are defined asthe tensor norms induced by the following mappings (see [20, p. 489]): (cid:0) ⊗ k,s ℓ ( B E ) , β k (cid:1) ։ (cid:0) ⊗ k,s E, \ β k / (cid:1) . (cid:0) ⊗ k,s E, /β k \ (cid:1) ֒ → (cid:0) ⊗ k,s ℓ ∞ ( B E ′ ) , β k (cid:1) . Recall that for a symmetric tensor norm β k , its dual tensor norm β ′ k is defined on finite dimensionalnormed spaces by (cid:0) ⊗ k,s M, β ′ k (cid:1) : = [ (cid:0) ⊗ k,s M ′ , β k (cid:1) ] ′ and then extended to Banach spaces so that it is finitely generated (see [25, 4.1]). OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 7
We say that β k is a natural symmetric tensor norm (of order k ) if β k is obtained from π k with a finitenumber of the operations \ / , / \ , ′ .For k ≥ , it is shown in [14] that there are exactly six non-equivalent natural tensor norms (note thatfor k = 2 there are only four). They can be arranged in the following diagram: π k ↑\ /π k \ / ր տ /π k \ \ ε k / տ ր / \ ε k / \↑ ε k where α k → γ k means that γ k dominates α k . There are no other dominations.Therefore, we have six “natural sequences” { α k } k of symmetric tensor norms, with their correspondingassociated polynomial ideals. If, as usual, we denote /π k \ by η k , we have η ′ k = \ ε k / , \ η k / = \ /π k \ / and /η ′ k \ = / \ ε k / \ .For the multiplicativity of the polynomial ideals associated to the natural symmetric tensor norms weneed the following: Lemma 2.9.
For each k , let α k be a finitely generated k -fold symmetric tensor and suppose there is aconstant c k,l > such that α k + l ( σ ( s ⊗ t )) ≤ c k,l α k ( s ) α l ( t ) for every s ∈ N k,sα k E ′ and t ∈ N l,sα l E ′ . Thenthe same inequality holds for the sequences { /α k \} k and {\ α k / } k .Proof. The argument used in [6, p.20] to show that { η k } k is a complemented sequence of seminorms canbe readily followed to show the statement for { /α k \} k . Indeed, if, for all k , we denote i k = ⊗ k i : (cid:0) ⊗ k,s E ′ , /α k \ (cid:1) ֒ → (cid:0) ⊗ k,s ℓ ∞ ( B E ′′ ) , α k (cid:1) , then /α k + l \ ( σ ( s ⊗ t )) = α k + l (cid:16) i k + l (cid:0) σ ( s ⊗ t ) (cid:1)(cid:17) = α k + l (cid:16) σ (cid:0) i k ( s ) ⊗ i l ( t ) (cid:1)(cid:17) ≤ c k,l α k ( i k ( s )) α l ( i l ( t )) = c k,l /α k \ ( s ) /α l \ ( t ) . On the other hand, if A maxk is the maximal ideal associated to α k , then by Proposition 2.6, if P ∈ A maxk ( E ) and Q ∈ A maxl ( E ) then P Q ∈ A maxk + l ( E ) and k P Q k A maxk + l ( E ) ≤ c k,l k P k A maxk ( E ) k Q k A maxl ( E ) . More-over, the identity \ α k / = ( /α ′ k \ ) ′ and the representation theorem for maximal polynomial ideals [27,Section 3.2] show that the maximal polynomial ideal B k associated to \ α k / at E is (cid:16) N k,s/α ′ k \ E (cid:17) ′ . Thus,since N k,s/α ′ k \ E is (isometrically) a subspace of N k,sα ′ k ℓ ∞ ( B E ′ ) , by the Hahn-Banach theorem B k ( E ) con-sists of all k -homogeneous polynomials on E which extend to α ′ k -continuous polynomials on ℓ ∞ ( B E ′ ) .That is, B k ( E ) = { P ∈ P k ( E ) : P extends to a polynomial ˜ P ∈ A maxk ( ℓ ∞ ( B E ′ )) } , and the norm of P in B k is given by the infimum of the A maxk -norms of these extensions. Then, it is easyto see that the product of two polynomials in B belongs to B with the same inequality of norms. UsingProposition 2.6 again, we obtain the desired result for {\ α k / } k . (cid:3) DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO
From Remark 2.7(b), a statement as in the previous lemma is true for the coherence conditions. As aconsequence, since π k and ε k are multiplicative, we can use the previous lemma and Proposition 2.6 toshow that: Theorem 2.10.
Let { α k } k be any of the natural sequences of symmetric tensor norms. Then the sequences { A maxk } k and { A mink } k of maximal and minimal ideals associated to { α k } k are multiplicative. Also, it is proved in [11] that the interpolation of multiplicative sequences is multiplicative.3.
Analytic structure on the spectrum
In [4] an analytic structure in the spectrum of H b ( U ) ( U an open subset of symmetrically regularBanach space) was given and it was shown that the functions in H b ( U ) have analytic extension to thespectrum. For the case of entire functions Dineen proved in [22, Section 6.3] that the extensions to thespectrum are actually of bounded type in each connected component of the spectrum.In this section we will show that it is possible to attach an analogous analytic structure to the spectrum M b A ( E ) of H b A ( E ) for a wide class of Banach spaces E and multiplicative sequences A . Then the spectrumturns out to a Riemann domain spread over E ′′ and, as in [4] or [22], each connected component of M b A ( E ) is an analytic copy of E ′′ . So we are able to define functions of the class H b A on each connected componentof M b A ( E ) . It follows that with an additional condition which is fulfilled for most of our examples we canprove that functions in H b A ( E ) extend to A -holomorphic functions of bounded type on each connectedcomponent of M b A ( E ) .For A a multiplicative sequence, let us consider the spectrum M b A ( E ) of the algebra H b A ( E ) (i.e. theset of continuous nonzero multiplicative functionals on H b A ). Since the inclusion H b A ( E ) ֒ → H b ( E ) iscontinuous, evaluations at points of E ′′ belong to M b A ( E ) . Therefore, δ z is a continuous homomorphismfor each z ∈ E ′′ and we can see E ′′ as a subset of M b A ( E ) .Also, given ϕ ∈ M b A ( E ) we can define an element π ( ϕ ) ∈ E ′′ by π ( ϕ )( γ ) = ϕ ( γ ) for every γ ∈ E ′ .Then the linear mapping π : M b A ( E ) → E ′′ ϕ ϕ | E ′ is a projection from M b A ( E ) onto E ′′ ⊂ M b A ( E ) . From the definition of π , for ϕ ∈ M b A ( E ) and γ ∈ E ′ we have ϕ ( γ N ) = ϕ ( γ ) N = (cid:16) π ( ϕ )( γ ) (cid:17) N = (cid:16) AB ( γ )( π ( ϕ )) (cid:17) N = AB ( γ N )( π ( ϕ )) . Thus, for every finite type polynomial P , ϕ ( P ) = AB ( P )( π ( ϕ )) = δ π ( ϕ ) ( P ) . As a consequence, we have the following:
Lemma 3.1.
Let A be a multiplicative sequence and E a Banach space such that, for every k , the finitetype k -homogeneous polynomials are dense in A k ( E ) . Then M b A ( E ) = E ′′ . Example 3.2.
Since the finite type polynomials are dense in any minimal ideal, if A is a multiplicativesequence of minimal ideals, then M b A ( E ) = E ′′ for any Banach space E . In particular, this happens forthe nuclear and the approximable polynomials, so M bN ( E ) = E ′′ and M ba ( E ) = E ′′ . OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 9
The Aron-Berner extension plays a crucial role in the analytic structure of M b ( E ) given in [4]. In orderto obtain a similar structure for our algebras, we need the polynomial ideals to have a good behavior withthese extensions. So let us introduce the following: Definition 3.3.
A sequence A of scalar valued ideals of polynomials is said to be AB -closed if thereexists a constant α > such that for each Banach space E , k ∈ N and P ∈ A k ( E ) we have that AB ( P ) belongs to A k ( E ′′ ) and k AB ( P ) k A k ( E ′′ ) ≤ α k k P k A k ( E ) , where AB denotes the Aron-Berner extension. Example 3.4.
The sequence A is known to be AB -closed with constant α = 1 in the following cases:continuous polynomials A = {P k } k (see [19]), integral polynomials A = {P kI } k (see [18]), extendiblepolynomials A = {P ke } k (see [8]), weakly continuous on bounded sets polynomials A = {P kw } k (see [31]),nuclear polynomials A = {P kN } k , approximable polynomials A = {P ka } k .In [13] it is shown that if A k is a maximal or a minimal ideal, then the Aron-Berner extension is anisometry from A k ( E ) into A k ( E ′′ ) , extending a well known result of Davie and Gamelin [19] and analogousresults for some particular polynomial ideals. Therefore, any sequence { A k } k of maximal (or minimal)polynomial ideals is AB -closed with constant α = 1 . Note that all the previous examples but A = {P kw } k are maximal or minimal, so they are covered by this result. Example 3.5.
It is easy to prove that if the sequences { A k } k and { A k } k are AB -closed with constants α and α respectively, then the interpolated sequence { A θk } k is AB -closed with constant α − θ α θ . Remark 3.6.
Note that as a consequence of the above definition, if A is coherent and AB -closed, foreach P ∈ A k ( E ) , j < k and z ∈ E ′′ , we have that AB ( P ) z k − j ∈ A j ( E ′′ ) and k AB ( P ) z k − j k A j ( E ′′ ) ≤ ( C k z k ) k − j α k k P k A k ( E ) .Moreover, since the A k ’s are ideals, if Q ∈ A j ( E ′′ ) then Q ◦ J E ∈ A j ( E ) and k Q ◦ J E k A j ( E ) ≤k Q k A j ( E ′′ ) . Therefore for each P ∈ A k ( E ) , AB ( P ) z k − j ◦ J E ∈ A j ( E ) and k AB ( P ) z k − j ◦ J E k A j ( E ) ≤ ( C k z k ) k − j α k k P k A k ( E ) , where J E denotes the canonical injenction of E into E ′′ .Also note that if f ∈ H b A ( E ) then AB ( f ) ∈ H b A ( E ′′ ) and p R ( AB ( f )) ≤ p αR ( f ) .We want now to define a topology on M b A ( E ) which makes ( M b A ( E ) , π ) into a Riemann domain. To dothis, we need first to prove that the translation is well define in spaces of holomorphic functions associatedto polynomial ideals. Lemma 3.7.
Let A be a multiplicative sequence, E a Banach space and x ∈ E . Then τ x : H b A ( E ) → H b A ( E ) f f ( x + · ) is a continuous operator. In particular, if ϕ ∈ H b A ( E ) ′ then ϕ ◦ τ x ∈ H b A ( E ) ′ and if ϕ ∈ M b A ( E ) then ϕ ◦ τ x ∈ M b A ( E ) .Proof. Take f = P ∞ k =0 P k ∈ H b A ( E ) and x ∈ E . Then P k ( x + y ) = P kj =0 (cid:0) kj (cid:1) ( P k ) x k − j ( y ) and τ x f = ∞ X k =0 k X j =0 (cid:18) kj (cid:19) ( P k ) x k − j . Using that the sequence is coherent it is easy to see that this series convergesabsolutely:(1) ∞ X k =0 k X j =0 (cid:18) kj (cid:19)(cid:13)(cid:13) ( P k ) x k − j (cid:13)(cid:13) A j ( E ) ≤ ∞ X k =0 (1 + C k x k ) k k P k k A k ( E ) = p C k x k ( f ) . So we can reverse the order of summation to obtain that d j τ x f (0) j ! = P ∞ k = j (cid:0) kj (cid:1) ( P k ) x k − j . Thus we obtainthat p R ( τ x f ) = ∞ X j =0 R j (cid:13)(cid:13)(cid:13) d j τ x f (0) j ! (cid:13)(cid:13)(cid:13) A j ( E ) ≤ ∞ X j =0 R j ∞ X k = j (cid:18) kj (cid:19)(cid:13)(cid:13) ( P k ) x k − j (cid:13)(cid:13) A j ( E ) = ∞ X k =0 k P k k A k ( E ) k X j =0 (cid:18) kj (cid:19) R j ( C k x k ) k − j ≤ p R + C k x k ( f ) . Therefore τ x f ∈ H b A ( E ) and τ x is continuous. (cid:3) If A is AB -closed and coherent and z ∈ E ′′ we can define ˜ τ z : H b A ( E ) → H b A ( E ) f τ z ( AB ( f )) ◦ J E . Remark 3.6 ensures that ˜ τ z is a (well-defined) continuous operator and p R (˜ τ z f ) ≤ p α ( R + C k z k ) ( f ) . Corollary 3.8.
Let A be an AB -closed and multiplicative sequence and let z ∈ E ′′ . Then ˜ τ z is a continuousoperator. Consequently, if ϕ ∈ H b A ( E ) ′ then ϕ ◦ ˜ τ z ∈ H b A ( E ) ′ and if ϕ ∈ M b A ( E ) then ϕ ◦ ˜ τ z ∈ M b A ( E ) . Note that π ( ϕ ◦ ˜ τ z )( γ ) = ϕ ◦ ˜ τ z ( γ ) = ϕ ( AB ( γ )( z + J E ( · )) = ϕ (1) z ( γ ) + ϕ ( γ ) = z ( γ ) + ϕ ( γ ) , and thus π ( ϕ ◦ ˜ τ z ) = z + π ( ϕ ) .A necessary condition to obtain the analytic structure of the spectrum of H b ( E ) is that the space E be symmetrically regular (i.e. the Arens extensions of every symmetric multilinear form are symmetric).In our case, to study the spectrum of H b A ( E ) , we need that the Arens extensions of ∨ P be symmetric forevery P in A k ( E ) (and for all k ). This happens, of course, if E is symmetrically regular, but also forarbitrary E if A k are good enough. So we define: Definition 3.9.
A sequence A is regular at E if, for every k and every P in A k ( E ) , we have thatevery Arens extension (that is, any extension by w ∗ -continuity in each variable in some order) of ∨ P aresymmetric. We say that the sequence A is regular if it is regular at E for every Banach space E . Example 3.10. (a) Any sequence of ideals contained in the ideals of approximable polynomials isregular. In particular, any sequence of minimal ideals is regular.(b) The sequences of integral [17, Proposition 2.14], extendible [17, Proposition 2.15] and weaklycontinuous [5] multilinear forms are regular.(c) If { α k } k is a sequence of projective symmetric tensor norms and A is a sequence of ideals associatedto { α k } k , then A is regular. Indeed, \ ε k / ≤ α k and thus α ′ k ≤ \ ε k / ′ = η k . Then, if we denote β k = α ′ k , since β k ≤ η k , every P ∈ (cid:16)N k,sβ k E (cid:17) ′ is extendible and, by (b), any Arens extension of ∨ P is symmetric. This says that the sequence of maximal ideals associated to { α k } k is regular andso is A .(d) Let { α k } k be a sequence of symmetric tensor norms and let A be the sequence of maximal poly-nomial ideals associated to { α k } k . If A is regular then is regular also any sequence of polynomialideals associated to { /α k \} k . Indeed, if we denote β k = α ′ k , for any P ∈ (cid:16) ⊗ k,s \ β k / E (cid:17) ′ we have that Q = P ◦ ⊗ k q ∈ (cid:16) ⊗ k,sβ k ℓ ( B E ) (cid:17) ′ = A k ( ℓ ( B E )) , where q is the metric projection ℓ ( B E ) ։ E . Weclaim that any Arens extension of ∨ P is symmetric. Indeed, let σ be a permutation of { , . . . , k } OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 11 and let e P , e Q the Arens extensions of ∨ P , ∨ Q defined by w ∗ -continuity in the order determined by σ . For j = 1 , . . . , k , take z j ∈ E ′′ and bounded nets ( x jλ j ) λ j ⊂ E such that x jλ j w ∗ → z j . Then e Q ( z , . . . , z k ) = lim λ σ (1) . . . lim λ σ ( k ) ∨ Q ( x λ , . . . , x kλ k ) = lim λ σ (1) . . . lim λ σ ( k ) ∨ P ( q ( x λ ) , . . . , ( x kλ k ))= e P ( q ′′ ( z ) , . . . , q ′′ ( z k )) . Since q ′′ is surjective and e Q is symmetric, we conclude that e P is symmetric.(e) As a consequence of (c) and (d) we obtain that if A is a sequence of polynomial ideals associatedwith any of the natural sequences (except for the case α k = ε k ) then A is regular.(f) If the sequences { A k } k and { A k } k are regular then the interpolated sequence { A θk } k is also regular(because each A θk is contained in A k + A k ).As in [4] it can be proved that, if A is regular at E then every evaluation at a point of E iv is in fact anevaluation at a point of E ′′ .The next two lemmas can be obtain just as in [22, Pages 428-430]. Lemma 3.11.
Let A be an AB -closed coherent sequence which is regular at a Banach space E . Then ˜ τ z ◦ ˜ τ w = ˜ τ z + w for every z, w ∈ E ′′ . Lemma 3.12.
Let A be an AB -closed multiplicative sequence which is regular at a Banach space E . Foreach ϕ ∈ M b A ( E ) and ε > define V ϕ,ε = { ϕ ◦ ˜ τ z : z ∈ E ′′ , k z k < ε } . Then (cid:8) V ϕ,ε : ϕ ∈ M b A ( E ) , ε > (cid:9) is the basis of a Hausdorff topology in M b A ( E ) . Proposition 3.13.
Let A be an AB -closed multiplicative sequence which is regular at a Banach space E . Then ( M b A ( E ) , π ) is a Riemann domain over E ′′ and each connected component of ( M b A ( E ) , π ) ishomeomorphic to E ′′ .Proof. With the topology defined in the above lemma, it is clear that for each ϕ ∈ M b A ( E ) and ε > , π | V ϕ,ε is an homeomorphism onto B E ′′ ( π ( ϕ ) , ε ) . Thus π : M b A ( E ) → E ′′ is a local homeomorphism. Notethat given ϕ ∈ M b A ( E ) , by Corollary 3.8, ϕ ◦ ˜ τ z is an homomorphism for each z ∈ E ′′ . Moreover, since π ( ϕ ◦ ˜ τ z ) = π ( ϕ ) + z it follows that π is an homeomorphism from S ( ϕ ) := { ϕ ◦ ˜ τ z : z ∈ E ′′ } to E ′′ andthus S ( ϕ ) is the connected component of ϕ in M b A ( E ) . (cid:3) Example 3.14. ( M b A ( E ) , π ) is a Riemann domain over E ′′ (and each connected component is homeo-morphic to E ′′ ) in the following cases:(a) A = {P kI } k or A = {P ke } k or, more generally, A the sequence of maximal polynomial idealsassociated to any of the natural sequences { α k } k (except for α k = ǫ k ) and E any Banach space.(b) A = {P kw } k and E any Banach space.(c) A any multiplicative sequence of maximal polynomial ideals and E symmetrically regular.(d) A = { A θk } k , with { A k } k and { A k } k any of the sequences of the examples (a) or (b) (or (c) and E symmetrically regular).Each function f ∈ H b A ( E ) can be extended via its Gelfand transform ˜ f to the spectrum M b A ( E ) , thatis ˜ f ( ϕ ) = ϕ ( f ) . Now that we have proved that M b A ( E ) is a Riemann domain, it is natural to ask if ˜ f is analytic in M b A ( E ) . Moreover, one can wonder if ˜ f preserve some of the properties of f in terms ofthe ideals A . Also, given ϕ ∈ H b A ( E ) ′ and f ∈ H b A ( E ) , Corollary 3.8 allows us to define a function on E ′′ by z ϕ ◦ ˜ τ z ( f ) . We will show in Theorem 3.18 that this function belongs to H b A ( E ′′ ) . This will allow us to conclude that the restriction of ˜ f to each connected component of M b A ( E ) is of A -holomorphic(Theorem 3.23 below).First, note that for ϕ ∈ H b A ( E ) ′ , there are constants c, r > such that | ϕ ( g ) | ≤ cp r ( g ) , for every g ∈ H b A ( E ) . In particular, | ϕ ( P ) | ≤ cp r ( P ) = cr k k P k A k ( E ) . As a consequence, we have(2) k ϕ | A k ( E ) k A k ( E ) ′ ≤ cr k , for every k ≥ .When a sequence of polynomial ideals is defined in both the scalar and vector valued case, one mayconsider the following property: “for every P ∈ A k ( E ) , the mapping x P x l belongs to the space ofvector-valued polynomials A l ( E ; A k − l ( E )) ”. This would mean that the differentials of a polynomial in A are also polynomials in A . Since we are dealing with scalar-valued polynomial ideals, we can consider asimilar property, which could be read as “the differentials of a polynomial in A are weakly in A ”. Moreprecisely, we have: Definition 3.15.
Let A be a coherent sequence of polynomial ideals and let E be a Banach space. Wesay that A is weakly differentiable if there exists a constant K such that, for l < k , P ∈ A k ( E ) and ϕ ∈ A k − l ( E ) ′ , the mapping x ϕ ( P x l ) belongs to A l ( E ) and (cid:13)(cid:13)(cid:13) x ϕ (cid:0) P x l (cid:1)(cid:13)(cid:13)(cid:13) A l ( E ) ≤ K k k ϕ k A k − l ( E ) ′ k P k A k ( E ) . Since d k − l P ( k − l )! ( x ) = (cid:0) kl (cid:1) P x l , the previous condition is equivalent to say that ϕ ◦ d k − l P ( k − l )! belongs to A l ( E ) and (cid:13)(cid:13)(cid:13) ϕ ◦ d k − l P ( k − l )! (cid:13)(cid:13)(cid:13) A l ( E ) ≤ (cid:18) kl (cid:19) K k k ϕ k A k − l ( E ) ′ k P k A k ( E ) . This is what we mean by saying that the differentials are weakly in A and what suggested our terminology.As the following proposition shows, there is some kind of duality between the properties of multiplica-tivity and weakly differentiability of a sequence. Proposition 3.16. ( i ) Let A = { A k } k be a weakly differentiable sequence. Then the sequence ofadjoint ideals { A ∗ k } k is multiplicative. ( ii ) Let A = { A k } k be a multiplicative sequence. Then the sequence of adjoint ideals { A ∗ k } k is weaklydifferentiable.In both cases the constants of multiplicativity and weakly differentiability are the same.Proof. ( i ) From [12, Proposition 5.1] we know that { A ∗ k } k is coherent. By Remark 2.7, it suffices to checkthat k P Q k A ∗ k + l ( M ) ≤ K k + l k P k A ∗ k ( M ) k Q k A ∗ l ( M ) for any finite dimensional Banach space M . Since M isfinite dimensional, A ∗ k ( M ) is just A k ( M ′ ) ′ . Take P ∈ A ∗ k ( M ) = A k ( M ′ ) ′ and Q ∈ A ∗ l ( M ) = A l ( M ′ ) ′ . For Ψ = P j x k + lj ∈ A k + l ( M ′ ) , we have h P Q, Ψ i = X j P ( x j ) Q ( x j ) = h Q, X j h P, x kj i x lj i = h Q, γ X j h P, x kj i x j ( γ ) l i = h Q, γ P (Ψ γ l ) i . Thus, since { A k } k is weakly differentiable, |h P Q, Ψ i| ≤ k Q k A ∗ l ( M ) k γ P (Ψ γ l ) k A l ( M ) ≤ k Q k A ∗ l ( M ) K k + l k P k A ∗ k ( M ) k Ψ k A k + l ( M ′ ) , which implies that k P Q k A ∗ k + l ( M ) ≤ K k + l k P k A ∗ k ( M ) k Q k A ∗ l ( M ) . OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 13 ( ii ) For each k , let α k be the finitely generated symmetric tensor norm associated to A k , so that forevery E , A ∗ k ( E ) = (cid:16) N k,sα k E (cid:17) ′ . By Proposition 2.6 and Remark 2.7 ( a ), the multiplicativity of A withconstant K implies that α k ( σ ( s ⊗ t )) ≤ K k α l ( s ) α k − l ( t ) , for every s ∈ N l,sα l E and t ∈ N k − l,sα k − l E .Take P ∈ A ∗ k ( E ) and ϕ ∈ A ∗ k − l ( E ) ′ , with k ϕ k = 1 . Define Q ( x ) = ϕ ( P x l ) . Note that Q is a well defined l -homogeneous polynomial since the sequence of adjoint ideals of a coherent sequence is again coherent[12, Proposition 5.1]. We have to prove that Q belongs to A ∗ l ( E ) = (cid:16) N l,sα l E (cid:17) ′ and that k Q k A ∗ l ( E ) ≤ K k k P k A ∗ k ( E ) . Take s = P j y lj ∈ N l,sα l E and ε > . Since ϕ ∈ (cid:16) N k − l,sα k − l E (cid:17) ′′ , by Goldstine theorem thereis some t = P i x k − li ∈ N k − l,sα k − l E , with α k − l ( t ) ≤ such that |h Q, s i| = | P j ϕ ( P y lj ) | ≤ |h P j P y lj , t i| + ε .Thus, |h Q, s i| ≤ |h X j P y lj , X i x k − li i| + ε = (cid:12)(cid:12) X i,j ∨ P ( x k − li , y lj ) (cid:12)(cid:12) + ε = (cid:12)(cid:12) h P, σ (cid:0) X i,j x k − li ⊗ y lj ) i (cid:12)(cid:12) + ε ≤ k P k A ∗ k ( E ) α k ( σ ( s ⊗ t )) + ε ≤ K k k P k A ∗ k ( E ) α l ( s ) α k − l ( t ) + ε. Since this is true for arbitrarily small ε and α k − l ( t ) ≤ , we conclude that k Q k A ∗ l ( E ) ≤ K k k P k A ∗ k ( E ) . (cid:3) Next corollary to Proposition 3.16 follows, for maximal ideals, from the equality A ∗∗ k = A k . For minimalideals is a consequence of Example 3.21 (f) below. Corollary 3.17.
Let { A k } k be sequence of maximal polynomial ideals or minimal polynomial ideals. Then { A k } k is weakly differentiable (multiplicative) if and only if { A ∗ k } k is multiplicative (weakly differentiable). If E is a Banach space and A is a weakly differentiable coherent sequence which is AB -closed (withconstant α ), it easily follows that the mapping E ′′ ∋ z ϕ ( AB ( P ) z l ◦ J E ) belongs to A l ( E ′′ ) and(3) (cid:13)(cid:13)(cid:13) z ϕ ( AB ( P ) z l ◦ J E ) (cid:13)(cid:13)(cid:13) A l ( E ′′ ) ≤ α k K k k ϕ k A k − l ( E ) ′ k P k A k ( E ) . Theorem 3.18.
Let A be an AB -closed weakly differentiable coherent sequence. For each ϕ ∈ ( H b A ( E )) ′ ,the following operator is well defined and continuous: ˜ T ϕ : H b A ( E ) → H b A ( E ′′ ) f ( z ϕ ◦ ˜ τ z ( f )) Proof.
Take f = P ∞ k =0 P k ∈ H b A ( E ) and z ∈ E ′′ . Then ϕ ◦ ˜ τ z ( f ) = P ∞ k =0 P kj =0 (cid:0) kj (cid:1) ϕ (cid:0) AB ( P k ) z j ◦ J E (cid:1) = P ∞ j =0 P ∞ k = j (cid:0) kj (cid:1) ϕ (cid:0) AB ( P k ) z j ◦ J E (cid:1) since using Remark 3.6 and inequality (2) it is easy to see that thisseries is absolutely convergent.Let Q l ( z ) = P ∞ k = l (cid:0) kl (cid:1) ϕ (cid:0) AB ( P k ) z l ◦ J E (cid:1) . Then ϕ ◦ ˜ τ z ( f ) = P ∞ l =0 Q l ( z ) . We will show that Q l belongsto A l ( E ′′ ) and that P ∞ l =0 Q l is in H b A ( E ′′ ) . To prove this it suffices to show that the series P ∞ k = l (cid:0) kl (cid:1)(cid:13)(cid:13)(cid:13) z ϕ (cid:0) AB ( P k ) z l ◦ J E (cid:1)(cid:13)(cid:13)(cid:13) A l ( E ′′ ) converges and that for every R > , the series P ∞ l =0 R l (cid:13)(cid:13)(cid:13) P ∞ k = l (cid:0) kl (cid:1) z ϕ (cid:0) AB ( P k ) z l ◦ J E (cid:1)(cid:13)(cid:13)(cid:13) A l ( E ′′ ) also converges. By inequality (3) we have ∞ X l =0 R l (cid:13)(cid:13)(cid:13) ∞ X k = l (cid:18) kl (cid:19) z ϕ (cid:0) AB ( P k ) z l ◦ J E (cid:1)(cid:13)(cid:13)(cid:13) A l ( E ′′ ) ≤ ∞ X l =0 R l ∞ X k = l (cid:18) kl (cid:19)(cid:13)(cid:13)(cid:13) z ϕ (cid:0) AB ( P k ) z l ◦ J E (cid:1)(cid:13)(cid:13)(cid:13) A l ( E ′′ ) ≤ ∞ X l =0 R l ∞ X k = l (cid:18) kl (cid:19) α k K k k ϕ | A k − l ( E ) k A k − l ( E ) ′ k P k k A k ( E ) ≤ c ∞ X k =0 α k k P k k A k ( E ) K k k X l =0 (cid:18) kl (cid:19) R l r k − l = cp αK ( R + r ) ( f ) , where in the last inequality we have used (2) and changed the order of summation. Therefore ˜ T ϕ ( f ) belongs to H b A ( E ′′ ) and p R ( ˜ T ϕ ( f )) ≤ cp αK ( R + r ) ( f ) , that is, ˜ T ϕ ∈ L ( H b A ( E ) , H b A ( E ′′ )) . (cid:3) With a similar proof to the above result one can prove:
Corollary 3.19.
Let A be a weakly differentiable multiplicative sequence. For ϕ ∈ H b A ( E ) ′ and f ∈ H b A ( E ) , we define ϕ ∗ f ( x ) = ϕ ◦ τ x ( f ) . Then we have ϕ ∗ f ∈ H b A ( E ) and the application T ϕ : H b A ( E ) → H b A ( E ) f ϕ ∗ f is a continuous linear operator.Moreover, if ψ ∈ M b A ( E ) we can define ϕ ∗ ψ ∈ H b A ( E ) ′ by ϕ ∗ ψ ( f ) = ψ ( ϕ ∗ f ) , and the application M ϕ : H b A ( E ) ′ → H b A ( E ) ′ ψ ψ ∗ ϕ is continuous. As usual, we call a continuous operator T : H b A ( E ) → H b A ( E ) that commutes with translations aconvolution operator. We have the following characterization: Corollary 3.20.
With the hypothesis and notation of the previous corollary, T : H b A ( E ) → H b A ( E ) is aconvolution operator if and only if there exist ϕ ∈ H b A ( E ) ′ such that T f = ϕ ∗ f for every f ∈ H b A ( E ) .Proof. If ϕ ∈ H b A ( E ) ′ then T ϕ ( f ) = ϕ ∗ f is continuous by the previous corollary and it is easily checked tobe a convolution operator. Conversely, let ϕ = T ◦ δ . Then T f ( x ) = τ x ( T f )(0) = T ( τ x f )(0) = ϕ ∗ f . (cid:3) Now we show that, again, our hypotheses are fulfilled by many sequences of polynomial ideals.
Example 3.21.
The following sequences are weakly differentiable:(a) A = {P k } k : if P ∈ P k ( E ) and ϕ ∈ P k − l ( E ) ′ then it is clear that x ϕ (cid:0) P x l (cid:1) ∈ P l ( E ) and (cid:13)(cid:13) x ϕ (cid:0) P x l (cid:1)(cid:13)(cid:13) P l ( E ) ≤ e l k ϕ k P k − l ( E ) ′ k P k P k ( E ) .(b) A = {P kI } k : this is a consequence of the above Proposition 3.16 ( ii ), since P kI = ( P k ) ∗ and {P k } is multiplicative with constant M = 1 .(c) A = {P ke } k : if P ∈ P ke ( E ) and ϕ ∈ P k − le ( E ) ′ then Q ( x ) = ϕ (cid:0) P x l (cid:1) is in P l ( E ) . Let E J ֒ → G and ˜ P an extension of P to G . Then ˜ Q ( y ) = ϕ (cid:0) ˜ P y l ◦ J (cid:1) is an extension of Q to G , and thus Q isextendible. Moreover, since | ˜ Q ( y ) | ≤ e l k y k l k ϕ kk ˜ P k , it follows that k Q k e ≤ e l k ϕ kk P k e .(d) A = {P kw } k : it is known (see [22, Proposition 2.6]) that if P ∈ P kw ( E ) then d k − l P is weaklycontinuous. Thus x ϕ (cid:0) P x l (cid:1) ∈ P lw ( E ) and has norm ≤ e l k ϕ k P k − l ( E ) ′ k P k P k ( E ) . OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 15 (e) A the sequence of maximal polynomial ideals associated to any of the natural sequences. Thisfollows from the multiplicativity of those sequences together with Proposition 3.16 ( ii ) , or fromthe previous examples (a) and (b) and Lemma 3.22 below.(f) If { A k } k is a weakly differentiable sequence and C is a normed operator ideal, then { A k ◦ C } k is weakly differentiable. In particular, the minimal hulls of the ideals in a weakly differentiablesequence form also a weakly differentiable sequence. Indeed, let P = RT ∈ A k ◦ C ( E ) , with T ∈ C ( E, E ) and R ∈ A k ( E ) . Take ϕ ∈ A k − l ◦ C ( E ) ′ and define ψ ∈ A k − l ( E ) ′ by ψ ( Q ) = ϕ ( QT ) .Note that k ψ k A k − l ( E ) ′ ≤ k ϕ k A k − l ◦ C ( E ) ′ k T k k − l C ( E,E ) . Clearly, ϕ ( P x l ) = ϕ ( R ( T x ) l T ) = ψ ( R ( T x ) l ) , thus x ϕ ( P x l ) = (cid:2) x ψ ( R x l ) (cid:3) ◦ T belongs to A l ◦ C ( E ) , because { A k } is weakly differentiableand T ∈ C . Moreover k x ϕ ( P x l ) k A l ◦ C ( E ) = k (cid:2) x ψ ( R x l ) (cid:3) ◦ T k A l ◦ C ( E ) ≤ k x ψ ( R x l ) k A l ( E ) k T k l C ( E,E ) ≤ K k k ψ k A k − l ( E ) ′ k R k A k ( E ) k T k l C ( E,E ) ≤ K k k ϕ k A k − l ◦ C ( E ) ′ k R k A k ( E ) k T k k C ( E,E ) . Since this is true for every factorization of P = RT , we conclude that k x ϕ ( P x l ) k A l ◦ C ( E ) ≤ K k k ϕ k A k − l ◦ C ( E ) ′ k P k A k ◦ C ( E ) . Lemma 3.22.
Let A be the sequence of maximal polynomial ideals associated to a sequence of symmetrictensor norms { α k } k . If A is weakly differentiable, then the same is true for the sequences of maximalpolynomial ideals associated to {\ α k / } k and to { /α k \} k .Proof. Let us denote β k = α ′ k . For A the sequence of maximal polynomial ideals associated to {\ α k / } k wehave that P belongs to A k ( E ) if and only if P ∈ (cid:16) ⊗ k,s/β k \ E (cid:17) ′ , and we can proceed just as we did with theideal of extendible polynomials (note that polynomials in A k ( E ) are those that extends to a β k -continuouspolynomial on ℓ ∞ ( B E ′ ) ).For the sequence of maximal ideals associated to { /α k \} k , P belongs to A k ( E ) if and only if e P = P ◦ q k belongs to (cid:16) ⊗ k,sβ k ℓ ( B E ) (cid:17) ′ = A k ( ℓ ( B E )) , where q k is the metric projection ⊗ k,sβ k ℓ ( B E ) ։ ⊗ k,s \ β k / E . Also,transposing q k we obtain a metric injection (cid:16) ⊗ k,s \ β k / E (cid:17) ′ ֒ → (cid:16) ⊗ k,sβ k ℓ ( B E ) (cid:17) ′ . If we take ϕ ∈ (cid:16) ⊗ k,s \ β k / E (cid:17) ′ , wecan choose a Hahn-Banach extension ψ of ϕ on ⊗ k,sβ k ℓ ( B E ) .Now, if Q ( x ) = ϕ (cid:0) P x l (cid:1) , we have Q ◦ q ( z ) = ϕ ( P q ( z ) l ) = ψ (( P ◦ q ) z l ) , which belongs to (cid:16) ⊗ k − l,sβ k ℓ ( B E ) (cid:17) ′ because A is weakly differentiable. But this means that Q belongs to A k − l ( E ) . (cid:3) In the previous proof, we only used that A is weakly differentiable in spaces of the form ℓ ∞ ( I ) (for {\ α k / } k ) and ℓ ( J ) (for { /α k \} k ), where I and J are some index sets.Now we are able to show that the extension of a function in H b A ( E ) to the spectrum “is A -holomorphicon each connected component”: Theorem 3.23.
Let E be a Banach space and A be an AB -closed multiplicative sequence which is regularat E and weakly differentiable. Then, given any function f ∈ H b A ( E ) and its extension ˜ f to M b A ( E ) , therestriction of ˜ f to each connected component of M b A ( E ) is a A -holomorphic function of bounded type.Proof. We have to show that for every ϕ ∈ M b A ( E ) , ˜ f ◦ (cid:0) π | S ( ϕ ) ) − ∈ H b A ( E ′′ ) . But note that S ( ϕ ) = { ϕ ◦ ˜ τ z : z ∈ E ′′ } and that (cid:0) π | S ( ϕ ) ) − ( z ) = ϕ ◦ ˜ τ z − π ( ϕ ) so ˜ f ◦ (cid:0) π | S ( ϕ ) ) − ( z ) = ϕ ◦ ˜ τ z − π ( ϕ ) ( f ) . That is, ˜ f ◦ (cid:0) π | S ( ϕ ) ) − = e T ϕ ◦ ˜ τ − π ( ϕ ) ( f ) which is in H b A ( E ′′ ) by Theorem 3.18. (cid:3) We can apply the last result in the following cases:
Example 3.24. ( M b A ( E ) , π ) is a Riemann domain over E ′′ and every function in H b A ( E ) extends to an A -holomorphic function of bounded type on each connected component of M b A ( E ) in the following cases:(a) A = {P k } k , and E is symmetrically regular (this is [22, Proposition 6.30]).(b) A = {P kI } k , for every Banach space E .(c) A = {P ke } k , for every Banach space E .(d) A = {P kw } k , for every Banach space E .(e) A a sequence of maximal polynomial ideals associated to any of the natural sequences but { ε k } k ,for every Banach space E .4. A Banach-Stone type result
Now we apply some of our results to obtain a Banach-Stone type theorem for algebras associated tomultiplicative sequences of polynomial ideals. We follow a procedure as in similar results in [16]. First,we have:
Lemma 4.1.
Let A and B be multiplicative sequences. Suppose that φ : H b A ( E ) → H b B ( F ) is a contin-uous multiplicative operator and define g : F ′′ → E ′′ by g ( z ) = π ( e δ z ◦ φ ) . Then, g is holomorphic and forevery γ ∈ E ′ , AB ( γ ) ◦ g = AB ( φγ ) . In particular, if the finite type polynomials are dense on A k ( E ) (forevery k ), then AB ( φf ) = AB ( f ) ◦ g for every f ∈ H b A ( E ) .Proof. Denote by θ φ : M b B ( F ) → M b A ( E ) the restriction of the transpose of φ . Then g is just thecomposition F ′′ e δ −→ M b B ( F ) θ φ −→ M b A ( E ) π −→ E ′′ . If we take z ∈ F ′′ and γ ∈ E ′ , then g ( z )( γ ) = e δ z ( φγ ) = AB ( φγ )( z ) . Thus g is weak*-holomorphic on F ′′ and therefore holomorphic (see for example[32, Example 8D]).If γ ∈ E ′ then AB ( γ )( g ( z )) = g ( z )( γ ) = AB ( φγ )( z ) . Since φ multiplicative and continuous, the lastassertion follows. (cid:3) Although it is hard for a Banach space E to satisfy that finite type polynomials are dense in H b ( E ) ( c and Tsirelson like spaces do, but no other classical Banach spaces), it is not so uncommon thatfinite type polynomials be dense in H b A ( E ) for certain sequences A and Banach spaces E . Besidesthose sequences where finite type polynomials are automatically dense (such as approximable or nuclearpolynomials), there are combination of ideals and Banach spaces that make finite type polynomials dense(see Example 4.3 below). Theorem 4.2. (a) Let A be an AB -closed multiplicative sequence such that finite type polynomials aredense in A k ( E ′′ ) . Then, H b A ( E ) and H b A ( F ) are topologically isomorphic algebras if and only if E ′ and F ′ are isomorphic. OLOMORPHIC FUNCTIONS AND POLYNOMIAL IDEALS ON BANACH SPACES 17 (b) Let A and B be multiplicative sequences, B also AB -closed. Suppose that finite type polynomialsare dense in A k ( E ) and on B k ( E ′′ ) for some Banach space E , for all k . If H b A ( E ) and H b B ( F ) aretopologically isomorphic algebras, then E ′ is isomorphic to F ′ .Proof. ( a ) ( b ) If E ′ and F ′ are isomorphic, so are E ′′ and F ′′ . Therefore, finite type polynomials are alsodense in A k ( F ′′ ) . Thus, A is regular both at E and F and we can follow the reasoning in [17, 30] to obtainthe desired isomorphism. The converse is a particular case of ( b ) . ( b ) Suppose that φ : H b A ( E ) → H b B ( F ) is an isomorphism. Let g : F ′′ → E ′′ and h : E ′′ → F ′′ be theapplications given by Lemma 4.1 for φ and φ − respectively. Then h ◦ g is the composition F ′′ e δ → M b B ( F ) θ φ → M b A ( E ) π → E ′′ e δ → M b A ( E ) θ φ − → M b B ( F ) π → F ′′ . Since M b A ( E ) = e δ ( E ′′ ) , it follows that h ◦ g = id F ′′ . Thus dh ( g (0)) ◦ dg (0) = id F ′′ and therefore F ′′ is isomorphic to a complemented subspace of E ′′ which implies that every polynomial in B k ( F ′′ ) isapproximable. Since B is AB -closed we can conclude that every polynomial in B k ( F ) is approximable (if P ∈ B k ( F ) then AB ( P ) ∈ B k ( F ′′ ) , thus AB ( P ) is approximable and therefore P is approximable). Now,since M b B ( F ) = e δ ( F ′′ ) , we can prove similarly that g ◦ h = id E ′′ , that is, h = g − , and differentiating at g (0) we obtain that E ′′ is isomorphic to F ′′ .Since every polynomial on B k ( F ) is approximable we have that φγ is weakly continuous on bounded setsfor every γ ∈ E ′ and then AB ( φγ ) is w ∗ -continuous on bounded sets. The identity g ( z )( γ ) = AB ( φγ )( z ) shown in Lemma 4.1 assures then that g is w ∗ - w ∗ -continuous on bounded sets. Similarly, g − is w ∗ - w ∗ -continuous on bounded sets. Moreover, applying [3, Lemma 2.1] to z g ( z )( γ ) , we obtain that dedifferential of g at any point is w ∗ - w ∗ -continuous (and analogously for g − ). Therefore, the isomorphismbetween E ′′ and F ′′ is the transpose of an isomorphism between F ′ and E ′ . (cid:3) Example 4.3.
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