Cyclicity of bicyclic operators

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http://france.elsevier.com/direct/CRASS1/

Functional Analysis

Cyclicity of bicyclic operators

Evgeny Abakumov

^a

, Aharon Atzmon

^b

, Sophie Grivaux

^c

aLaboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France

bSchool of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel

cLaboratoire Paul-Painlevé, UMR 8524, Université des sciences et technologies de Lille, Cité Scientifique, bâtiment M2, 59655 Villeneuve d’Ascq cedex, France

Received 15 January 2007; accepted 30 January 2007 Available online 21 March 2007

Presented by Gilles Pisier

Abstract

We study cyclicity of injective operators on separable Banach spaces which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. Our results apply in particular to the shift operator acting on the weighted spaces of sequences²_ω(Z). We also prove completeness results of translates in certain Banach spaces of functions onR.To cite this article: E. Abakumov et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Résumé

Cyclicité de certains opérateurs bicycliques.Nous étudions la cyclicité de certains opérateurs bicycliques agissant sur des espaces de Banach séparables. Nos résultats s’appliquent en particulier aux opérateurs de décalage sur les espaces de suites pondérés ²_ω(Z). Nous prouvons également des résultats concernant la complétude des translatés d’une fonction dans certains espaces de Banach de fonctions surR.Pour citer cet article : E. Abakumov et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Version française abrégée

SoitX un espace de Banach séparable complexe, etT un opérateur borné surX injectif et bicyclique : il existe un vecteurx₀∈R^∞(T )=

n1Im(Tⁿ)tel que les vecteursTⁿx₀,n∈Z, engendrent un sous-espace dense deX. Le but de cette Note est de présenter une méthode permettant de déduire queT est en fait cyclique lorsque les normes Tⁿx₀,n∈Z, vérifient certaines conditions de croissance. Voici notre résultat principal :

Théorème 0.1. SoitT ∈B(X)un opérateur injectif admettant un vecteur bicycliquex₀. Supposons qu’il existe un entierk0et une suite sous-multiplicative(ρ(n))_n₀de nombres strictement positifs telle que

E-mail addresses:[email protected] (E. Abakumov), [email protected] (A. Atzmon), [email protected] (S. Grivaux).

doi:10.1016/j.crma.2007.02.008

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n→+∞lim

log⁺ρ(n)

√n =0 et

(1) Tⁿx0 =O(n^k)lorsquentend vers l’infini; (2) T⁻ⁿx0 =O(ρ(n))lorsquentend vers l’infini.

Si le spectre ponctuelσp(T^∗)deT^∗ne contient pas le cercle unité, alorsT est cyclique.

Ce théorème permet par exemple de retrouver le fait que le décalage à droiteS:(a_n)_n_∈Z→(a_n₋₁)_n_∈Zest cyclique sur^p(Z)pour toutp >1. Plus généralement :

Théorème 0.2. SoitXun espace de Banach dont le dual est séparable, etUun opérateur unitaire surX(c’est-à-dire une isométrie surjective). SiUest bicyclique, alorsUest cyclique.

Une deuxième conséquence du Théorème 0.1 concerne le décalage à droite agissant sur l’espace de suites ²_ω(Z)=

(a_n)_n_∈Z∈C^Z;(a_n)

²_ω(Z)=

n∈Z

|a_n|²ω(n)² 1/2

<+∞

,

oùωest une suite sous-multiplicative de nombres strictement positifs. C’est une question ouverte que de caracté- riser les poidsωayant la propriété que S est cyclique sur²_ω(Z). La première partie du théorème suivant s’ensuit directement du Théorème 0.1 :

Théorème 0.3. Supposons queω(n)=1pour toutn0 : (1) silog⁺ω(n)/√

ntend vers0lorsquentend vers l’infini, alorsSest cyclique sur²_ω(Z);

(2) si la série _n>0logω(−n)/n² diverge et siωest suffisamment régulier pour que le Théorème de Volberg sur l’intégrale logarithmique s’applique, alorsSn’est pas cyclique sur²_ω(Z).

Nous formulons la conjecture suivante :

Conjecture 0.1.Siω(−n)=e^√ⁿpour toutn >0etω(n)=1pour toutn0, alorsSn’est pas cyclique sur²_ω(Z).

Herrero pose dans [7] la question suivante : siωest un poids quelconque, l’un des deux opérateursSouS^∗doit-il nécessairement être cyclique sur²_ω(Z)? Nous prouvons :

Théorème 0.4. La réponse à la question ci-dessus est positive lorsque

n∈Z

logSⁿ

1+n² <+∞.

Soitw:R→ [1,+∞)un poids sous-multiplicatif mesurable surR. AlorsL¹_w(R)est une algèbre de Banach pour la convolution. La méthode employée pour prouver le Théorème 0.1 permet de retrouver simplement un résultat de [2], qui affirme que siΛest un sous-ensemble deRdont le rayon de Beurling–MalliavinR(Λ)est infini, et siwest un poids non quasianalytique, il existe une fonctionfappartenant à l’algèbre de BeurlingL¹_w(R)telle que l’ensemble des translatésf (· −λ),λ∈Λ, est dense dansL¹_w(R). Voici un autre résultat qui peut être obtenu par la même méthode : Théorème 0.5.SoitXun espace de Banach séparable,πune représentation fortement continue deRsurX, etΛun sous-ensemble deRtel queR(Λ)= +∞. Siπ est non quasianalytique et cyclique, il existe un vecteurx∈Xtel que la famille desπ(λ)x,λ∈Λ, engendre un sous-espace dense deX.

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1. Cyclicity of bicyclic operators on Banach spaces

Our aim in this Note is to present conditions on a bounded operatorT acting on a separable Banach spaceXwhich imply that ifT isbicyclic, thenT must in fact becyclic. Proofs will be presented elsewhere.

Definition 1.1.LetT be an injective operator onX, and defineR^∞(T )=

n1Ran(Tⁿ)to be the intersection of the ranges of all the operatorsTⁿ. We say thatT isbicyclicif there exists a vectorx0∈R^∞(T )(a bicyclic vector forT) such that the linear span of the vectorsTⁿx₀,n∈Z, is dense inX. Recall thatT is said to becyclicif there exists a vectorx∈Xsuch that the linear span of the vectorsTⁿx,n0, is dense inX.

Recall that a sequence(ρ(n))_n_∈_A,A=Z⁺orZ, is submultiplicative ifρ(n+m)ρ(n)ρ(m)forn, m∈A.

Theorem 1.2.LetT be a bounded injective operator onXwhich admits a bicyclic vectorx0∈R^∞(T ). Suppose that the following conditions are satisfied:there exist a non-negative integerkand a submultiplicative sequence(ρ(n))n0

of positive numbers with

n→+∞lim

log⁺ρ(n)

√n =0 such that

(1) Tⁿx₀ =O(n^k)asngoes to infinity;

(2) T⁻ⁿx₀ =O(ρ(n))asngoes to infinity.

If the point spectrumσ_p(T^∗)ofT^∗does not include the unit circleT= {λ∈C; |λ| =1}, thenT is cyclic.

In particular ifT is an invertible operator this yields:

Theorem 1.3.LetT be an invertible bicyclic operator onXsuch that

(1) there exists a non-negative integerksuch thatTⁿ =O(n^k)asngoes to infinity;

(2) logT⁻ⁿ/√

ntends to0asngoes to infinity.

Ifσp(T^∗)does not include the unit circle, thenT is cyclic.

Here is another consequence of Theorem 1.2 concerning surjective isometries of Banach spaces:

Theorem 1.4.LetUbe a unitary operator on a separable Banach spaceX.

(1) ifUis bicyclic andσ_p(U^∗)does not includeT, thenUis cyclic;

(2) ifXhas separable dual andUis bicyclic, thenUis cyclic.

For unitary operators acting on a Hilbert space this result is known and follows from a more general result [3] on cyclicity of normal operators. Theorem 1.4 applies in particular to all the unweighted bilateral shiftsS_p:(a_n)_n_∈Z→ (a_n₋₁)_n_∈Zon the spaces^p(Z)forp >1. This result forS_pwas proved independently by A. Olevski˘ı in 1998 by a different method (unpublished manuscript).

2. Cyclicity ofSonAρ,0(T)

Letρbe a submultiplicative weight withρ(n)1 for everyn∈Z. The main argument of the proof of Theorem 1.2 involves a study of the multiplication operator by e^iθ, denoted byS, on the Banach space

A_ρ(T)=

f ∈C(T); fρ=

n∈Z

f (n)ρ(n) <ˆ +∞

.

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Definition 2.1.A submultiplicative sequenceρ:Z→ [1,+∞)such that

n∈Z

logρ(n) 1+n² <+∞

will be called aBeurling sequence.

Ifρ is a Beurling sequence, it is known thatA_ρ(T)is a regular Banach algebra, so it contains functions with arbitrarily small support. The closed idealA_ρ,0(T)defined as the closure inA_ρ(T)of the set of the functions which vanish identically in a neighborhood of the point 1 (which is clearly invariant under the action ofS) appears as a natural space on which to study the cyclicity properties ofS:

Theorem 2.2.Ifρis a Beurling sequence, the operatorSacting onAρ,0(T)is cyclic.

The proof of Theorem 2.2 relies on a Baire Category argument: we prove, using the fact thatA_ρ(T)is a regular Banach algebra, that given two non-empty open subsets ofA_ρ,0(T)there exists an analytic trigonometric polynomial psuch thatp(S)(U )∩V is non-empty. For more on this Baire Category method and its applications to universality questions, we refer the reader to the survey [5].

To deduce Theorem 1.2 from Theorem 2.2, the key argument is the following result of [1]: ifρ(n)=(n+1)^k for n0 and limn→+∞log⁺ρ(−n)/√

n=0,thenA_ρ,0(T)coincides with the closed ideal generated by the function (e^iθ−1)^k⁺¹, i.e. the closure of the set

Y_ρ,0^(k)(T)=

e^iθ−1k+1

f; f∈A_ρ(T) .

3. Cyclicity of shifts on weighted Hilbert spaces of sequences onZ

The main motivation for Theorem 1.2 is the study of cyclicity properties of a particularly interesting class of operators: the shift operator on weighted Hilbert or Banach spaces of sequences onZ. Letωbe a positive weight on Zsuch that sup_n_∈Zω(n+1)/ω(n) <+∞.Then the shift operatorS:(a_n)_n_∈Z→(a_n₋₁)_n_∈Zacting on the weighted ²-space

²_ω(Z)=

(a_n)_n_∈Z∈C^Z; (a_n)

²_ω(Z)=

n∈Z

|a_n|²ω(n)² 1/2

<+∞

is bounded on²_ω(Z). This operatorS has been the subject of various investigations since the 60’s (see [9,8]), concerning in particular non-trivial invariant and bi-invariant subspaces. The description of the invariant subspaces of the operatorS acting on ²_ω(Z)is closely connected to the description of the cyclic vectors ofS. One of the main applications of Theorem 1.2 is to obtain a partial answer to the following open problem:

Problem 3.1.Characterize the weightsωsuch thatSis cyclic on²_ω(Z).

This problem is mentioned in [6], and partial results were obtained by Herrero [7], who proved thatSis cyclic if its point spectrum is non-empty, and not cyclic if the point spectrum ofS^∗ is non-empty. Shifts on various Banach spaces of functions and hyperdistributions are studied extensively in [8]. SinceS is always bicyclic in the sense of Definition 1.1 with bicyclic vector(δ0,n)n∈Z, Theorem 1.2 can be applied here:

Theorem 3.2.Letωbe a positive weight such thatS is bounded on²_ω(Z). Suppose that there exist a non-negative integerkand a submultiplicative sequence(ρ(n))_n₀of positive numbers with

n→+∞lim

log⁺ρ(n)

√n =0 such that

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(1) ω(n)=O(n^k)asngoes to infinity;

(2) ω(−n)=O(ρ(n))asngoes to infinity.

ThenSis cyclic on²_ω(Z)if and only if

n∈Z

1

ω(n)² = +∞.

For instance if for someα∈R,ω(n)=(|n| +1)^α for everyn∈Z, thenSis cyclic on²_ω(Z)if and only ifα1/2.

The same conclusion holds true ifω(n)=(n+1)^α forn0 andω(−n)=exp(√

n/log(n+1))forn >0. Although Theorem 3.2 does not yield a complete characterization of the weights (withω(n)=1 for everyn0, for instance) such thatSis cyclic on²_ω(Z), it leads us to a reasonable conjecture concerning this characterization. If we specialize to the case whereω(n)=1 forn0, then Theorem 3.2 tells us thatSis cyclic on²_ω(Z)ifω(−n)grows “slower that e^√ⁿ”. The case whereω(−n)=e^√ⁿforn >0 appears as a limit case, and our conjecture is that the shift is not cyclic for this weight:

Conjecture 3.3.Ifω0is the weight defined byω0(n)=1forn0andω0(−n)=e^√ⁿforn >0, thenSis not cyclic on²_ω

0(Z).

On the other hand, it is not difficult to see that whenω(n)=1 forn0,ω(n)1 for everyn∈Z,

n<0

logω(n) 1+n² = +∞

andωis sufficiently regular,Sis not cyclic on²_ω(Z): this follows directly from the known characterization of cyclic vectors for the shift on²(Z)and Volberg’s theorem on the logarithmic integral [10]. This holds true for instance if ω(n)=1 for everyn0 andω(−n)=exp(n/log(n+1))for everyn >0. But ifω(n)=exp(n/log(|n| +2))for everyn∈Z, we do not know whetherSis cyclic or not on²_ω(Z).

4. On a question of Herrero

The following question was posed by Herrero in [7]:

Question 4.1.Ifωis a positive weight onZ, must eitherSorS^∗be cyclic on²_ω(Z)?

The following general theorem for operators on Banach spaces, as well as some easy considerations on supercyclic operators, yields a positive answer to Question 4.1 in the non-quasianalytic case:

Theorem 4.2.LetXbe a complex separable Banach space andT a bounded injective operator onXwhich admits a bicyclic vectorx₀∈R^∞(T ). Suppose that the following conditions are satisfied:there exist a non-negative integerk and a Beurling sequenceρsuch that

(1) Tⁿx₀ =O(ρ(n))as|n|goes to infinity;

(2) Tⁿx₀.T⁻ⁿx₀ =O(n^k)asngoes to infinity.

Ifσ_p(T^∗)does not include the unit circle, thenT is cyclic.

Theorem 4.3.LetSbe a bounded invertible shift on²_ω(Z). Suppose that there exists a Beurling sequenceρsuch that ω(n)=O(ρ(n)). Then eitherSorS^∗is cyclic on²_ω(Z). This applies in particular if the operatorSsatisfies

n∈Z

logSⁿ

1+n² <+∞.

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5. Completeness of translates

The Baire Category method combined with the regularity of the Banach algebra applies very well to the study of completeness of translates of functions in some Banach algebras of functions overRwhich are invariant by all translations. Herew:R→ [1,+∞)will be a positive measurable function (weight) such thatw(t+s)w(t )w(s) for everyt, s∈R. Then

L¹_w(R)=

f; fL¹_w(R)=

R

f (t )w(t )dt <+∞

is a Banach algebra with respect to convolution, and all the translation operatorsf →f_λ=f (· −λ),λ∈R, are continuous onL¹_w(R). IfΛis a subset ofR,Λis calledgeneratinginL¹_w(R)if there exists a functionf ∈L¹_w(R) such that the translatesf_λ,λ∈Λ, span a dense subspace ofL¹_w(R). The main result of [4] is thatΛis generating inL¹(R)if and only ifR(Λ)= +∞, whereR(Λ)is the Beurling–Malliavin radius of the setΛ. The result of [4]

is extended to non-quasianalytic Beurling algebrasL¹_w(R)by Blank in [2]. We give a straightforward proof of this result, using only the fact that ifwis non-quasianalytic,L¹_w(R)is a regular Banach algebra:

Theorem 5.1.[2] If the weightwsatisfies

R

logw(t )

1+t² <+∞,

thenΛis generating forL¹_w(R)if and only ifR(Λ)= +∞.

We deduce from Theorem 5.1 the following result concerning cyclic representations ofRon Banach spaces:

Theorem 5.2.Letπbe a non-quasianalytic strongly continuous representation ofRon a separable Banach spaceX, andΛa subset ofRsuch thatR(Λ)= +∞. Ifπ is cyclic, then there exists a vectorx∈Xsuch that the linear span of the vectorsπ(λ)x,λ∈Λ, is dense inX.

References

[1] A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980) 27–63.

[2] N. Blank, Generating sets for Beurling algebras, J. Approx. Theory 140 (2006) 61–70.

[3] J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94.

[4] J. Bruna, A. Olevski˘ı, A. Ulanovski˘ı, Completeness inL¹(R)of discrete translates, Rev. Mat. Iberoamericana 22 (2006) 1–16.

[5] K.G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999) 345–381.

[6] R. Gellar, D. Herrero, Hyperinvariant subspaces of bilateral weighted shifts, Indiana Univ. Math. J. 23 (1973/74) 771–790.

[7] D. Herrero, Eigenvectors and cyclic vectors for bilateral weighted shifts, Rev. Un. Mat. Argentina 26 (1972/73) 24–41.

[8] N. Nikolski˘ı, Selected problems of weighted approximation and spectral analysis, Proc. Steklov Inst. Math. 120 (1974). English translation:

Amer. Math. Soc., Providence, RI, 1976.

[9] A. Shields, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, in: Math. Surveys, vol. 13, Amer. Math. Soc., Providence, RI, 1974, pp. 49–128.

[10] A. Volberg, Summability of the logarithm of a quasianalytic function, Dokl. Akad. Nauk SSSR 265 (1982) 1297–1302. English translation:

Soviet Math. Dokl. 26 (1982) 238–243.