Some improvements of the Katznelson-Tzafriri theorem on Hilbert space
aa r X i v : . [ m a t h . F A ] O c t SOME IMPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM ON HILBERT SPACE
DAVID SEIFERT
Abstract.
This paper extends two recent improvements in the Hilbert spacesetting of the well-known Katznelson-Tzafriri theorem by establishing botha version of the result valid for bounded representations of a large class ofabelian semigroups and a quantified version for contractive representations.The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certainunbounded representations. Introduction
In [18], Katznelson and Tzafriri proved that, given a power-bounded operator T on a complex Banach space X , k T n ( I − T ) k → n → ∞ if (and onlyif) σ ( T ) ∩ T ⊂ { } , where σ ( T ) denotes the spectrum of T and T is the unitcircle. Given a sequence a ∈ ℓ ( Z + ), define b a ( λ ) := P ∞ n =0 a ( n ) λ n ( | λ | ≤
1) and b a ( T ) := P ∞ n =0 a ( n ) T n , which is a bounded linear operator on X . Katznelson andTzafriri also showed, in the same paper, that(1.1) lim n →∞ k T n b a ( T ) k = 0provided there exists a sequence ( a n ) in ℓ ( Z + ) such that each c a n vanishes onan open neighbourhood of σ ( T ) ∩ T and k a − a n k → n → ∞ . This resulthas itself subsequently been extended, first to the case of C -semigroups (see[15, Th´eor`eme III.4] and [28, Theorem 3.2]) and later to more general semigrouprepresentations (see [6, Theorem 4.3] and [28]).These results are optimal in various senses (see [11, Section 5]), but improve-ments are possible when X is assumed to be a Hilbert space. It is shown in [14,Corollary 2.12], for instance, that the weaker (and necessary) condition that b a vanish on σ ( T ) ∩ T is sufficient for (1.1) to hold, at least when T is a contraction;see [19, Proposition 1.6] for a slightly more general result. This result has in turnbeen improved upon in two recent papers. In [20], L´eka has extended this resultto power-bounded operators on Hilbert space, and Zarrabi in [30] has shown thatfor contractions, and likewise for pairs of commuting contractions and for con-tractive C -semigroups, the limit appearing in (1.1) is given, more generally, bysup {| b a ( λ ) | : λ ∈ σ ( T ) ∩ T } or the appropriate analogue; for related results see [2],[3], [9], [10] and [22]. Date : 27 March 2013.2010
Mathematics Subject Classification.
Primary: 47D03; secondary: 43A45, 43A46, 47A35.
The purpose of this paper is to improve both L´eka’s and Zarrabi’s versions ofthe Katznelson-Tzafriri theorem by extending them to representations of a signif-icantly larger class of semigroups. The main result, Theorem 3.1, is a Katznelson-Tzafriri type theorem which holds for bounded (as opposed to contractive) repre-sentations and thus includes both [20, Theorem 2.1] and various results containedas special cases in [30]. The main implication of Theorem 3.1 is proved firstvia a certain ergodic condition as in [20] and then by a more direct argument.The second method does not involve the ergodic condition and, as is shown inTheorem 4.2, leads naturally to an extension of Zarrabi’s quantified results forcontractive representations. Section 5, finally, contains a brief exposition of animproved version of the Katznelson-Tzafriri theorem for individual orbits. Firstof all, though, Section 2 sets out the necessary preliminary material.2.
Preliminaries
The setting throughout, even when not stated explicitly, will be that of anabelian semigroup S contained in a locally compact abelian group G satisfying G = S − S . The Haar measure on G is denoted by µ , and it is assumed that S is Haar-measurable and hence itself becomes a measure space with respect to therestriction of µ . Assume furthermore that the interior S ◦ of S (in the topologyinduced by G ) is non-empty. The semigroup S becomes a directed set under therelation (cid:23) , where s (cid:23) t for s, t ∈ S whenever s − t ∈ S ∪ { } ; this makes itpossible to speak of limits as s → ∞ through S . The dual group of G , consistingof all continuous bounded characters χ : G → C , is denoted by Γ, the set ofcontinuous bounded characters on S by S ∗ . It follows from the assumption that S spans G that the subset of S ∗ of characters taking values in the unit circle T can (and will throughout) be identified with Γ. Two important examples of theabove are the (semi)groups Z (+) with counting measure and R (+) with Lebesguemeasure. Here S ∗ can be identified in a natural way with { λ ∈ C : | λ | ≤ } and { λ ∈ C : Re λ ≤ } , respectively, and the dual group Γ is T and i R in each case.For Ω = G or S, let L (Ω) denote the algebra (under convolution) of functions a : Ω → C that are integrable with respect to (the restriction of) Haar measureand, given a ∈ L (Ω), define its Fourier transform by b a ( χ ) := Z Ω a ( s ) χ ( s ) d µ ( s ) , where χ is an element of Γ or S ∗ , as appropriate. Given a closed subset Λ of Γ,define J Λ := { a ∈ L ( G ) : b a ≡ χ ∈ Λ } and K Λ := { a ∈ L ( G ) : b a ( χ ) = 0 for all χ ∈ Λ } . An element of L ( G ) is said to be of spectralsynthesis with respect to Λ if it lies in the closure of J Λ . Since K Λ is closed, anysuch function must be an element of K Λ . If K Λ coincides with the closure of J Λ ,the set Λ is said to be of spectral synthesis .For any closed subset Λ of Γ, the map W Λ : a b a | Λ is a well-defined con-tractive algebra homomorphism from L ( G ) into C (Λ) whose kernel is K Λ and MPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM 3 whose range, by the Stone-Weierstrass theorem, is dense in C (Λ); see [26, The-orem 1.2.4]. Since K Λ ( G ) is a closed ideal of L ( G ), W Λ induces a well-definedinjective algebra homomorphism U Λ : L ( G ) /K Λ ( G ) → C (Λ) which, by the In-verse Mapping Theorem, is an isomorphism precisely when it is surjective. Sinceits range is dense in C (Λ), this is the case if and only if the map is an isomorphicembedding (which in turn is equivalent to the dual operator U ′ Λ : M (Λ) → K ⊥ Λ being either a surjection or an isomorphic embedding, where M (Λ) denotes theset of complex-valued regular measures on Λ which have finite total variation; seefor instance [26, Appendix C11]). When these conditions are satisfied, Λ is said tobe a Helson set , and the quantity α (Λ) := k U − k is known as its Helson constant .Since U Λ is contractive, α (Λ) ≥ { b a : a ∈ L ( S ) } separates pointsboth from each other and from zero and, furthermore, that the interior S ◦ is densein S . For further details and discussion of these conditions, see for instance [6].These assumptions ensure, in particular, that there exists a net (Ω α ), known asa Følner net , of compact, measurable, non-null subsets of S satisfyinglim α →∞ µ (cid:0) Ω α △ (Ω α + s ) (cid:1) µ (Ω α ) = 0 , uniformly for s in compact subsets of S .Given a Banach space X and Ω = G or S for S and G as above, a representation of Ω on X is a strongly continuous homomorphism T : Ω → B ( X ) which, if 0 ∈ Ω,satisfies T (0) = I. The representation is said to be bounded if sup {k T ( s ) k : s ∈ Ω } < ∞ and in this case, given a ∈ L (Ω), the operator b a ( T ) ∈ B ( X ) is defined,for each x ∈ X , by b a ( T ) x := Z Ω a ( s ) T ( s ) x d µ ( s ) . Given a bounded representation T of G on a Banach space X and a closedsubspace Λ of the dual group Γ, the corresponding spectral subspace M T (Λ) isdefined as M T (Λ) := \ a ∈ J Λ Ker b a ( T ) , the (Arveson) spectrum Sp( T ) of T asSp( T ) := \ (cid:8) Λ ⊂ Γ : Λ is closed and M T (Λ) = X (cid:9) . Thus the spectrum is a closed subset of Γ, and it is shown in [24, Theorems 8.1.4and 8.1.12], respectively, that Sp( T ) is non-empty whenever X is non-trivial andthat it is compact if and only if T is continuous with respect to the norm topologyon B ( X ). If T is a representation by isometries, this notion of spectrum coincideswith the finite L-spectrum of [21, Section 5.2]. By [24, Proposition 8.1.9], fur-thermore, Sp( T ) has the alternative descriptionSp( T ) = (cid:8) χ ∈ Γ : | b a ( χ ) | ≤ k b a ( T ) k for all a ∈ L ( G ) (cid:9) . DAVID SEIFERT
Accordingly, given a bounded representation T of a semigroup S on a Banachspace X , the spectrum Sp( T ) of T is defined asSp( T ) := (cid:8) χ ∈ S ∗ : | b a ( χ ) | ≤ k b a ( T ) k for all a ∈ L ( S ) (cid:9) , and the unitary spectrum of T is given by Sp u ( T ) := Sp( T ) ∩ Γ; see [6] for details.In the examples mentioned above, bounded semigroup representations correspondto a single power-bounded operator T ∈ B ( X ) if S = Z + and to a bounded C -semigroup if S = R + . The spectrum is given by σ ( T ) and σ ( A ), respectively,where A denotes the generator of the semigroup.3. A general Katznelson-Tzafriri type result
The aim of this section is to prove the following generalisation of [20, Theo-rem 2.1].
Theorem 3.1.
Let T be a bounded representation of a semigroup S on a Hilbertspace X , and suppose that a ∈ L ( S ) . Then the following are equivalent:(i) b a ( χ ) = 0 for every χ ∈ Sp u ( T ); (ii) Given any Følner net (Ω α ) for S and any χ ∈ Sp u ( T ) , (3.1) lim α →∞ µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ( s ) T ( s ) b a ( T ) d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) = 0; (iii) k T ( s ) b a ( T ) k → as s → ∞ . Remark 3.2.
In [20, Theorem 2.1], conditions (ii) and (iii) above are presentedin a slightly more general form, with the operator b a ( T ) replaced by an arbitrary Q ∈ B ( X ) that commutes with the representation. The presentation here isrestricted to the case Q = b a ( T ) purely for simplicity.The proof of this result will be broken up into a number of separate steps, allof which correspond to some part of the proof of [20, Theorem 2.1] given by L´ekabut typically with some modifications to accommodate the more general settingin which the representation need not be norm continuous. The following lemmaconstitutes the main step towards proving that (i) = ⇒ (ii); it corresponds to [20,Lemma 2.2]. Note that the Hilbert space assumption is not required for this partof the argument. Lemma 3.3.
Let T be a bounded representation of a semigroup S on a Banachspace X , and let a ∈ L ( S ) . Then, for all χ ∈ Γ , lim α →∞ µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ( s ) T ( s ) (cid:0)b a ( T ) − b a ( χ ) (cid:1) d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 , where (Ω α ) is any Følner net for S and the integral is taken in the strong sense. Proof . With a ∈ L ( S ) and χ ∈ Γ fixed, let x ∈ X have unit norm and set I x ( α ) := 1 µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ( s ) T ( s ) (cid:0)b a ( T ) x − b a ( χ ) x (cid:1) d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) . MPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM 5
Then, by a simple application of Fubini’s theorem, I x ( α ) ≤ M Z S µ (cid:0) Ω α △ (Ω α + s ) (cid:1) µ (Ω α ) | a ( s ) | d µ ( s ) , where M := sup {k T ( s ) k : s ∈ S } . Let ε >
0. Since a ∈ L ( S ), there exists acompact subset K of S such that R S \ K | a ( s ) | d µ ( s ) < ε/ M. Defining ξ K ( α ) := sup (cid:26) µ (Ω α △ (Ω α + s )) µ (Ω α ) : s ∈ K (cid:27) , it follows from the definition of a Følner net that ξ K ( α ) → α → ∞ . Since I x ( α ) ≤ M k a k ξ K ( α ) + 2 M Z S \ K | a ( s ) | d µ ( s ) ,I x ( α ) < ε for all sufficiently large α , and the result follows. (cid:3) Corollary 3.4.
Let T be a bounded representation of a semigroup S on a Banachspace X , and let χ ∈ Γ . Suppose that a ∈ L ( S ) is such that b a ( χ ) = 0 . Then (3.1) holds for any Følner net (Ω α ) for S . The next result is an important step towards establishing the implication(ii) = ⇒ (iii) in Theorem 3.1 and should be compared with [20, Lemma 2.4]. Proposition 3.5.
Let S be a semigroup and T a representation of a group G = S − S by unitary operators on a Hilbert space X . Suppose that a ∈ L ( G ) andthat, for each χ ∈ Sp( T ) , lim α →∞ µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ( s ) T ( s ) b a ( T ) d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 , where (Ω α ) is any Følner net for S . Then b a ( T ) = 0 . Proof . Writing B (Γ) for the set of Borel subsets of the dual group Γ, let E : B (Γ) → B ( X ) denote the spectral measure associated with T (see [24, Theo-rem 8.3.2]) and, for s ∈ G and Λ ∈ B (Γ), let T Λ ( s ) := T ( s ) E (Λ). Then T Λ ( s ) := Z Λ χ ( s ) d E ( χ ) , the integral being taken in the weak sense, and, by Fubini’s theorem,(3.2) b b ( T Λ ) = Z Λ b b ( χ ) d E ( χ )for all b ∈ L ( G ). Thus if Λ ∈ B (Γ) is closed and b ∈ J Λ ( G ), then b b ( T Λ ) = 0,and it follows that M T Λ (Λ) = X , so that Sp( T Λ ) ⊂ Λ. Choosing Λ ∈ B (Γ) to becompact ensures that the representation T Λ of G on X is norm continuous.Set Q := b a ( T ) and, for a given compact subset Λ of Sp( T ), define Q Λ := QE (Λ), noting that Q Λ is normal and that Q Λ → Q in the weak (and indeedthe strong) operator topology as Λ approaches Sp( T ) through compact subsets.Furthermore, let A Λ denote the commutative unital C ∗ -algebra generated by { Q Λ , Q ∗ Λ } ∪ { T Λ ( s ) : s ∈ G } , and let ∆( A Λ ) denote its character space. WriteΦ Λ : A Λ → C (∆( A Λ )) for the Gelfand transform of A Λ , which is an isometric DAVID SEIFERT ∗ -isomorphism, and consider the map χ ξ : G → C \{ } given, for ξ ∈ ∆( A Λ ) and s ∈ G , by χ ξ ( s ) := Φ Λ ( T Λ ( s ))( ξ ) . Since the representation T Λ is norm continuous, χ ξ is a continuous group homomorphism, and the fact that each ξ ∈ ∆( A Λ )is a bounded linear functional on A Λ with k ξ k = | ξ ( E (Λ)) | = 1 implies that | χ ξ ( s ) | ≤ s ∈ G . Hence χ ξ ∈ Γ. Moreover, if b ∈ L ( G ), then (cid:12)(cid:12)b b ( χ ξ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ξ (cid:18)Z G b ( s ) T Λ ( s ) d µ ( s ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)b b ( T Λ ) (cid:13)(cid:13) , which is to say that χ ξ ∈ Sp( T Λ ), and hence χ ξ ∈ Sp( T ). Let g Λ := Φ Λ ( Q Λ ).Then | g Λ ( ξ ) | = 1 µ (Ω α ) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω α | χ ξ ( s ) | g Λ ( ξ ) d µ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13) Φ Λ (cid:18)Z Ω α χ ξ ( s ) T Λ ( s ) Q Λ d µ ( s ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = 1 µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ξ ( s ) T Λ ( s ) Q Λ d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ µ (Ω α ) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω α χ ξ ( s ) T ( s ) Q d µ ( s ) (cid:13)(cid:13)(cid:13)(cid:13) , for any ξ ∈ ∆( A Λ ) and letting α → ∞ shows that g Λ = 0 . Since Φ Λ is anisometry, it follows that Q Λ = 0, and allowing Λ to approach Sp( T ) throughcompact subsets gives Q = 0, as required. (cid:3) Remark 3.6.
The result remains true when b a ( T ) is replaced by any Q ∈ B ( X )which commutes with T . If Q is normal, this follows from the same argument asabove; the general case can then be obtained by considering the operator Q ∗ Q ;see also [20, Lemma 2.4].Propositions 3.7 and 3.8 below correspond in essence to the two main stages inthe proof of [20, Theorem 2.1] and show, via an intermediate result for the strongoperator topology, that (ii) = ⇒ (iii) in Theorem 3.1. Proposition 3.7.
Let T be a bounded representation of a semigroup S on aHilbert space X , and let a ∈ L ( S ) . Suppose that, for some Følner net (Ω α ) for S , (3.1) holds for all χ ∈ Sp u ( T ) . Then, for all x ∈ X , k T ( s ) b a ( T ) x k → as s → ∞ . Proof . Fix a Banach limit φ on L ∞ ( S ). A construction analogous to [6, Propo-sition 3.1] and [19, Section 1] shows that there exist a Hilbert space X φ , a rep-resentation T φ of S on X φ by isometries with Sp( T φ ) ⊂ Sp( T ) and an oper-ator π φ : X → X φ with the following properties: π φ is bounded with norm k π φ k ≤ sup {k T ( s ) k : s ∈ S } , k π φ ( x ) k = φ ( k T ( · ) x k ) for all x ∈ X , Ran π φ isdense in X φ , Ker π φ = { x ∈ X : k T ( s ) x k → s → ∞} and π φ T ( s ) = T φ ( s ) π φ for all s ∈ S (so π φ is an intertwining operator ). In particular, for any oper-ator Q ∈ B ( X ) that commutes with T , the operator Q φ ∈ B ( X φ ) defined by π φ Q = Q φ π φ satisfes k Q φ k ≤ k Q k . By a construction analogous to [4, Propo-sition 2.1] (see also [6, Proposition 3.2], [8], [13] and [17]), there exist a further MPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM 7
Hilbert space Y φ , a representation T G of the group G = S − S by unitary op-erators on Y φ with Sp( T G ) = Sp u ( T φ ) and an isometric intertwining operator π G : X φ → Y φ such that { T G ( − s ) π G ( x ) : s ∈ S, x ∈ X φ } is dense in Y φ . Thelatter implies, in particular, that k Q G k = k Q φ k for all Q φ ∈ B ( X φ ) and all Q G ∈ B ( X G ) which commute with T G and satisfy π G Q φ = Q G π G . Thus it ispossible to assume, sacrificing only the density condition on the range of theintertwining operator, that T φ itself is in fact a representation of G by unitaryoperators on X φ .Now, given χ ∈ Sp( T φ ), define operators Q α ∈ B ( X ) and Q φ,α ∈ B ( X φ ) as(3.3) Q α := 1 µ (Ω α ) Z Ω α χ ( s ) T ( s ) b a ( T ) d µ ( s )and Q φ,α := 1 µ (Ω α ) Z Ω α χ ( s ) T φ ( s ) b a ( T φ ) d µ ( s ) . Then π φ Q α = Q φ,α π φ , from which it follows that k Q φ,α k ≤ k Q α k . In particular, k Q φ,α k → α → ∞ . Identifying L ( S ) in the natural way with a subset of L ( G ), it follows from Proposition 3.5 applied to T φ , X φ and a that b a ( T φ ) = 0.In particular, π φ ( b a ( T ) x ) = b a ( T φ ) π φ ( x ) = 0 for any x ∈ X , so the result followsfrom the description of Ker π φ . (cid:3) Proposition 3.8.
Let T be a bounded representation of a semigroup S on aHilbert space X , and let a ∈ L ( S ) . Suppose that, for some Følner net (Ω α ) for S , equation (3.1) is satisfied for all χ ∈ Sp u ( T ) . Then k T ( s ) b a ( T ) k → as s → ∞ . Proof . Suppose, for the sake of contradiction, that (3.1) holds for all χ ∈ Sp u ( T )and some Følner net (Ω α ) but that there exist ε > s β ) in S , withindexing set A say, such that s β → ∞ as β → ∞ and, for some suitable sequence( y β ) of unit vectors in X , k T ( s β ) b a ( T ) y β k ≥ ε for all β ∈ A . Letting M :=sup {k T ( s ) k : s ∈ S } , it follows that k T ( s ) b a ( T ) y β k ≥ εM − whenever s β − s ∈ S .Fix t ∈ S ◦ and, essentially as in [26, Section 1.1.8], let b ∈ L ( S ) satisfy k b k = 1and k a ∗ b − a t k < ε/ M , where a t ∈ L ( S ) is given by a t ( s ) = a ( s − t ) if s − t ∈ S and a t ( s ) = 0 otherwise. Now, by a construction similar those containedin [16], [23] and [25], there exist(a) a Banach space X ∞ A , which is contained in the set ℓ ∞ ( A ; X ) of X -valuednets indexed by A and contains all nets of the form ( b c ( T ) x α ) with c ∈ L ( S ) and ( x α ) ∈ ℓ ∞ ( A ; X ), and a bounded representation T ∞ A of S on X ∞ A with Sp( T ∞ A ) = Sp( T );(b) a Hilbert space X A , a bounded representation T A of S on X A with Sp( T A ) ⊂ Sp( T ∞ A ) and a surjective intertwining operator π A : X ∞ A → X A which iscontractive and such that k π A ( x α ) k is given, for each ( x α ) ∈ X ∞ A , by thelimit of the net ( k x α k ) along some ultrafilter on A which contains the filtergenerated by the sets { α ∈ A : α (cid:23) β } with β ∈ A . DAVID SEIFERT
Note, in particular, that Sp( T A ) ⊂ Sp( T ). Consider the element ( x β ) of X ∞ A ,where x β := b b ( T ) y β . Then, writing c := a ∗ b − a t , k T A ( s ) b a ( T A ) π A ( x β ) k = k π A T ∞ A ( s ) b a ( T ∞ A )( x β ) k = (cid:13)(cid:13) π A (cid:0) T ( s ) d a ∗ b ( T ) y β (cid:1)(cid:13)(cid:13) ≥ (cid:13)(cid:13) π A (cid:0) T ( s + t ) b a ( T ) y β (cid:1)(cid:13)(cid:13) − (cid:13)(cid:13) π A (cid:0) T ( s ) b c ( T ) y β (cid:1)(cid:13)(cid:13) ≥ lim inf β →∞ k T ( s + t ) b a ( T ) y β k − M k c k for all s ∈ S , where the last line follows from the definition of the norm on X A .Thus k T A ( s ) b a ( T A ) π A ( x β ) k ≥ ε/ M for all s ∈ S .Fix χ ∈ Sp u ( T A ) and define the operators Q ∞ A,α ∈ B ( X ∞ A ) and Q A,α ∈ B ( X A )as Q ∞ A,α := 1 µ (Ω α ) Z Ω α χ ( s ) T ∞ A ( s ) b a ( T ∞ A ) d µ ( s )and Q A,α := 1 µ (Ω α ) Z Ω α χ ( s ) T A ( s ) b a ( T A ) d µ ( s ) . Then π A Q ∞ A,α = Q A,α π A , so the properties of π A and the fact that Q ∞ A,α acts on X ∞ A by entrywise application of the operator Q α , as defined in equation (3.3),imply that k Q A,α k ≤ k Q α k . Hence k Q A,α k → α → ∞ , and Proposition 3.7applied to T A and X A leads to the required contradiction. (cid:3) Corollary 3.4 and Proposition 3.8 together prove the implications (i) = ⇒ (ii) = ⇒ (iii) of Theorem 3.1. The following simple lemma, which follows immediatelyfrom the definition of the spectrum of a semigroup representation T along withthe observation that b a s ( T ) = T ( s ) b a ( T ) for all a ∈ L ( S ) and s ∈ S , shows that(iii) = ⇒ (i), thus completing the proof of the main result. Lemma 3.9.
Let T be a bounded representation of a semigroup S on a Banachspace X , and let a ∈ L ( S ) . Then | b a ( χ ) | ≤ k T ( s ) b a ( T ) k for all χ ∈ Sp u ( T ) and all s ∈ S . Remark 3.10.
There is a direct proof of the implication (iii) = ⇒ (ii) in Theo-rem 3.1. Indeed, if T is a bounded representation of a semigroup S on any Banachspace X , if a ∈ L ( S ) and if (Ω α ) is any Følner net for S , then k Q α k ≤ sup {k T ( s ) b a ( T ) k : s (cid:23) t } + M k a k µ (cid:0) Ω α △ (Ω α + t ) (cid:1) µ (Ω α )for any t ∈ S , where Q α is as in (3.3) and M = sup {k T ( s ) k : s ∈ S } . Hence(iii) = ⇒ (ii) by definition of a Følner net. Moreover, it is possible, at least inspecial cases, to show directly that (ii) = ⇒ (i). When S = Z + , this follows fromCorollary 3.4 and the uniform ergodic theorem (see [20, Corollary 2.3]), and asimilar argument works when S = R + .If one is interested in establishing only the equivalence of statements (i) and(iii) of Theorem 3.1, there is a shorter argument which may be of independentinterest. Recall the classical fact that, given a representation T of a group G MPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM 9 by isometries on a Banach space X , one has b a ( T ) = 0 for all a ∈ L ( G ) whichare of spectral synthesis with respect to Sp( T ). This is a simple consequence ofthe definition of the spectrum (see also [12, Chapter 8], [21, Chapter 5] and [27,Lemma 2.4.3]) and is used (together with constructions analogous to those usedin the proof of Proposition 3.7) in [6, Theorem 4.3] to derive a general form ofthe Katznelson-Tzafriri theorem on Banach space. Corollary 3.13 below, whichis an improved version of this result when X is a Hilbert space, makes it possibleto obtain the implication (i) = ⇒ (iii) of Theorem 3.1 by an analogous argumentwhich bypasses Proposition 3.5. It is a special case of the following more generalstatement. Proposition 3.11.
Let T be a representation of a group G by unitary operatorson a Hilbert space X , and let a ∈ L ( G ) . Then k b a ( T ) k = sup {| b a ( χ ) | : χ ∈ Sp( T ) } . Proof . Let A T denote the norm closure in B ( X ) of { b b ( T ) : g ∈ L ( G ) } . Then A T is a commutative C ∗ -algebra and hence, writing ∆( A T ) for the character space of A T , the Gelfand transform Φ : A T → C (∆( A T )) is an isometric ∗ -isomorphism.By [6, Proposition 2.4], the map sending a character χ ∈ Sp( T ) to the character ξ χ on A T defined, on the dense subspace { b b ( T ) : g ∈ L ( G ) } , by ξ χ ( b b ( T )) := b b ( χ )is a bijection, and hence k b a ( T ) k = k Φ( b a ( T )) k ∞ = sup {| b a ( χ ) (cid:12)(cid:12) : χ ∈ Sp( T ) } . (cid:3) Remark 3.12.
This result can also be proved using (3.2) with Λ = Sp( T ). Corollary 3.13.
Let T be a representation of a group G by unitary operators ona Hilbert space X with spectrum Λ := Sp( T ) , and let a ∈ L ( G ) . Then b a ( T ) = 0 if and only if a ∈ K Λ . Remark 3.14.
This follows also from Corollary 3.4 and Proposition 3.5 togetherwith Lemma 3.9.4.
Quantified results for contractive representations
The purpose of this section is to study the limit of k T ( s ) b a ( T ) k as s → ∞ in thecase where T is a contractive representation of a semigroup S on a Hilbert space X and a is an element of L ( S ) whose Fourier transform b a does not necessarilyvanish on the unitary spectrum Sp u ( T ) of T . Theorem 4.2 below constitutesan important step towards this aim and can be viewed as a sharper version of[3, Proposition 5.5] which holds on general Banach space. It follows from thefollowing result for individual orbits. Proposition 4.1.
Let T be a representation of a semigroup S by contractions ona Hilbert space X with unitary spectrum Λ := Sp u ( T ) , and let a ∈ L ( S ) . Then lim s →∞ k T ( s ) b a ( T ) x k ≤ k a + K Λ kk x k for all x ∈ X . Proof . Fix any Banach limit φ on L ∞ ( S ) and let X φ , T φ and π φ be as in the proofof Proposition 3.7. Then, since T is contractive, k π φ ( x ) k = lim s →∞ k T ( s ) x k for all x ∈ X and it is possible, as before, to assume that T φ is in fact a representationof the group G = S − S on X φ by unitary operators. It follows from Corollary 3.13that b b ( T φ ) = 0 for all b ∈ K Λ . Hence k b a ( T φ ) k ≤ k a − b k for any such b , whichimplies that k b a ( T φ ) k ≤ k a + K Λ ( G ) k . Thus, given any x ∈ X ,lim s →∞ k T ( s ) b a ( T ) x k = k π φ ( b a ( T ) x ) k = k b a ( T φ ) π φ ( x ) k≤ k a + K Λ ( G ) kk π φ ( x ) k , and the result follows since π φ is a contraction. (cid:3) Theorem 4.2.
Let T be a representation of S by contractions on a Hilbert space X with unitary spectrum Λ := Sp u ( T ) , and let a ∈ L ( S ) . Then lim s →∞ k T ( s ) b a ( T ) k ≤ k a + K Λ k . Proof . Suppose not. Then there exist ε > s β ) in S , with indexingset A , say, and satisfying s β → ∞ as β → ∞ , as well as a net of unit vectors( y β ) in X such that k T ( s β ) b a ( T ) y β k ≥ k a + K Λ k + ε for all β ∈ A . In particular, k T ( s ) b a ( T ) y β k ≥ k a + K Λ k + ε whenever s β − s ∈ S .Let X A , X ∞ A , T A and π A be as in the proof of Proposition 3.8, choose, for afixed t ∈ S ◦ , b ∈ L ( S ) such that k b k = 1 and k a ∗ b − a t k < ε/
2, and againdefine ( x β ) ∈ X ∞ A by setting x β := b b ( T ) y β . It then follows from a calculationanalogous to the one in the proof of Proposition 3.8 that k T A ( s ) b a ( T A ) π A ( x β ) k > k a + K Λ k + ε/ s ∈ S . However, applying Proposition 4.1 to the contractiverepresentation T A of S on X A , and using the fact that k π A ( x β ) k ≤ k b b ( T ) k ≤ , leads to lim s →∞ k T A ( s ) b a ( T A ) π A ( x β ) k ≤ k a + K Λ A kk π A ( x β ) k ≤ k a + K Λ k , where Λ A := Sp u ( T A ) ⊂ Λ. This gives the required contradiction. (cid:3)
The final result is a special instance Theorem 4.2 which applies to representa-tions whose unitary spectrum is a Helson set. It is a simple consequence of thedefinition of a Helson set and should be compared with the results in [30, Sec-tion 5], which hold on Banach space but in addition assume the unitary spectrumto be of spectral synthesis.
Corollary 4.3.
Let T be a representation of a semigroup S by contractions ona Hilbert space X , and let a ∈ L ( S ) . Suppose that the unitary spectrum Λ :=Sp u ( T ) of T is a Helson set. Then sup {| b a ( χ ) | : χ ∈ Λ } ≤ lim s →∞ k T ( s ) b a ( T ) k ≤ α (Λ) sup {| b a ( χ ) | : χ ∈ Λ } . Remark 4.4.
The first inequality holds irrespective of whether the unitary spec-trum is a Helson set, and indeed of whether X is a Hilbert space. It is animmediate consequence of Lemma 3.9. MPROVEMENTS OF THE KATZNELSON-TZAFRIRI THEOREM 11 Local results
This final section gives a brief account of how some of the aforementionedimprovements of the Katznelson-Tzafriri theorem on Hilbert space carry over toorbitwise, or ‘local’, versions of the result. More specifically, the aim is to obtainspectral conditions which ensure, given a bounded representation T of a semigroupon a Hilbert space X and an element a of L ( S ), that k T ( s ) b a ( T ) x k → s → ∞ for a particular point x ∈ X . Such results are of particular interest in the contextof C -semigroups, where orbits correspond to solutions of the associated Cauchyproblem, and they have been studied for instance in [5], [7, Section 4] and [19].The notion of spectrum that is most appropriate to this context goes backto [1]. Consider a bounded representation T of a semigroup S on some Banachspace X , and let x ∈ X be given. A character χ ∈ Γ will be said to be locallyregular at x if there exist n ∈ N , a , . . . , a n ∈ L ( S ), a neighbourhood Ω of thepoint λ := ( b a ( χ ) , . . . , c a n ( χ )) in C n and holomorphic functions g , . . . , g n : Ω → X such that P nk =0 ( λ k − b a k ( T )) g k ( λ ) = x for all λ = ( λ , . . . , λ n ) ∈ Ω. The unitary local (Albrecht) spectrum Sp u ( T ; x ) of T at x is then defined as the setof all characters χ ∈ Γ which fail to be locally regular at x . It is easy to seethat Sp u ( T ; x ) ⊂ Sp u ( T ) for each x ∈ X . For further details on the unitarylocal spectrum and its relation to other natural notions of local spectrum, see [7,Section 4].The main result of this section is the following theorem, which improves [5,Theorem 5.1] in the Hilbert space setting. Theorem 5.1.
Let T be a bounded representation of a semigroup S on a Hilbertspace X . Furthermore, let x ∈ X and a ∈ L ( S ) . Then k T ( s ) b a ( T ) x k → as s → ∞ provided b a ( χ ) = 0 for all χ ∈ Sp u ( T ; x ) . Proof . Fix a Banach limit φ on L ∞ ( S ) and let X φ , T φ and π φ be as in theproof of Proposition 3.7, so that T φ may again be assumed to be a representationof the group G = S − S on X φ by unitary operators. By [7, Proposition 5.1],Sp u ( T φ ; π φ ( x )) ⊂ Sp u ( T ; x ). Moreover, writing X x for the closed linear span ofthe set { T φ ( s ) π φ ( x ) : s ∈ G } in X φ and T x for the representation of G on X x obtained by restricting T φ , [7, Theorem 4.5] gives Sp( T x ) = Sp u ( T φ ; π φ ( x )) andhence Sp( T x ) ⊂ Sp u ( T ; x ). Thus the assumption on a implies that b a ( χ ) = 0 forall χ ∈ Sp( T x ), from which it follows by Corollary 3.13 that b a ( T x ) = 0. Hence π φ ( b a ( T ) x ) = b a ( T φ ) π φ ( x ) = 0, which is to say b a ( T ) x ∈ Ker π φ , as required. (cid:3) Theorem 5.1 in fact holds even for certain unbounded representations providedthe growth of the norm is sufficiently slow and regular. Given a semigroup S , ameasurable function w : S → [1 , ∞ ) is said to be a weight if it is bounded oncompact subsets of S and satisfies w ( s + t ) ≤ w ( s ) w ( t ) for all s, t ∈ S . Givena weight w and a representation T of S on a Banach space X which satisfies k T ( s ) k ≤ w ( s ) for all s ∈ S , it is possible, essentially by replacing any occurrenceof L ( S ) with the Beurling algebra L w ( S ), to define the modified unitary local(Albrecht) spectrum Sp w u ( T ; x ) for any point x ∈ X ; see [8] and [21] for details. An argument entirely analogous to the proof of Theorem 5.1, but this time usingthe full strength of the results in [7], then leads to the following result. Here theadditional regularity assumption on the weight w ensures that the representationcorresponding to T φ in the above proof is again isometric; see [7, Proposition 3.1].Similar non-local results may be found in [7, Theorem 3.4] and [29, Theorem 8]. Theorem 5.2.
Let T be a representation of a semigroup S on a Hilbert space X and suppose that T is dominated by a weight w such that, for every t ∈ S , w ( s ) − w ( s + t ) → as s → ∞ . Furthermore, let x ∈ X and suppose that a ∈ L w ( S ) is such that b a ( χ ) = 0 for all χ ∈ Sp w u ( T ; x ) . Then k T ( s ) b a ( T ) x k = o ( w ( s )) as s → ∞ . Acknowledgements
The author is grateful to Professor C.J.K. Batty for his guidance, to the EPSRCfor its financial support, and to the anonymous referee, whose careful reading ofan earlier version led to various minor improvements.
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