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# [

Tanja Eisner    Sophie Grivaux
###### Abstract

A strictly increasing sequence of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator  on a separable Hilbert space such that , the set of eigenvalues of modulus of is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.

Hilbertian Jamison sequences and rigid dynamical systems]Hilbertian Jamison sequences and rigid dynamical systems

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2000 Mathematics Subject Classification.  — 47A10, 37A05, 37A50, 47A10, 47B37.\@footnotetextKey words and phrases.  — Linear dynamical systems, partially power-bounded operators, point spectrum of operators, hypercyclicity, weak mixing and rigid dynamical systems, topologically rigid dynamical systems..\@footnotetextThis work was partially supported by ANR-Projet Blanc DYNOP, the European Social Fund and the Ministry of Science, Research and the Arts Baden-Württemberg..\par\par\par\par We are concerned in this paper with the study of certain dynamical systems, in particular linear dynamical systems. Our main aim is the study of rigidity sequences for weakly mixing dynamical systems on measure spaces, and we present tractable conditions on the sequence which imply that it is (or not) a rigidity sequence. Our conditions on the sequence come in part from the study of the so-called Jamison sequences, which appear in the description of the relationship between partial power-boundedness of an operator on a separable Banach space and the size of its unimodular point spectrum. \parLet us now describe our results more precisely. \par Let be a separable infinite-dimensional complex Banach space, and let be a bounded operator on . We are first going to study here the relationship between the behavior of the sequence of the norms of the powers of , and the size of the unimodular point spectrum , i.e. the set of eigenvalues of of modulus . It is known since an old result of Jamison [20] that a slow growth of makes small, and vice-versa. More precisely, the result of [20] states that if is power-bounded, i.e. , then is at most countable. For a sample of the kind of results which can be obtained in the other direction, let us mention the following result of Nikolskii [29]: if is a bounded operator on a separable Hilbert space such that has positive Lebesgue measure, then the series is convergent. This has been generalized by Ransford in the paper [34], which renewed the interest in these matters. In particular Ransford started to investigate in [34] the influence of partial power-boundedness of an operator on the size of its unimodular point spectrum. Let us recall the following definition: \par

###### Definition 1.1.  —

Let be an increasing sequence of positive integers, and a bounded linear operator on the space . We say that is partially power-bounded with respect to  if .

\par

In view of the result of Jamison, it was natural to investigate whether the partial power-boundedness of with respect to  implies that is at most countable. It was shown in [35] by Ransford and Roginskaya that it is not the case: if for instance, there exist a separable Banach space and such that is finite while is uncountable. This question was investigated further in [2] and [3], where the following definition was introduced: \par\par

###### Definition 1.2.  —

Let be an increasing sequence of integers. We say that is a Jamison sequence if for any separable Banach space and any bounded operator  on , is at most countable as soon as is partially power-bounded with respect to .

\par

Whether is a Jamison sequence or not depends of course on features of the sequence such as its growth, its arithmetical properties, etc. A complete characterization of Jamison sequences was obtained in [3]. It is formulated as follows: \par

###### Theorem 1.3.  —

Let be an increasing sequence of integers with . The following assertions are equivalent:

• is a Jamison sequence;

• there exists a positive real number such that for every ,

 supk≥0|λnk−1|≥ε.
\par

Many examples of Jamison and non-Jamison sequences were obtained in [2] and [3]. Among the examples of non-Jamison sequences, let us mention the sequences such that tends to infinity, or such that divides for each and . Saying that is not a Jamison sequence means that there exists a separable Banach space and such that and is uncountable. But the space may well be extremely complicated: in the proof of Theorem 1.3, the space is obtained by a rather involved renorming of a classical space such as for instance. This is a drawback in applications, and this is why it was investigated in [2] under which conditions on the sequence it was possible to construct partially power-bounded operators with respect to  with uncountable unimodular point spectrum on a Hilbert space. It was proved in [2] that if the series is convergent, there exists a bounded operator  on a separable Hilbert space such that and is uncountable. But this left open the characterization of Hilbertian Jamison sequences. \par

###### Definition 1.4.  —

We say that is a Hilbertian Jamison sequence if for any bounded operator  on a separable infinite-dimensional complex Hilbert space which is partially power-bounded with respect to , is at most countable.

\par

Obviously a Jamison sequence is a Hilbertian Jamison sequence. Our first goal in this paper is to prove the somewhat surprising fact that the converse is true: \par

###### Theorem 1.5.  —

Let be an increasing sequence of integers. Then is a Hilbertian Jamison sequence if and only if it is a Jamison sequence.

\par

Contrary to the proofs of [2] and [3], the proof of Theorem 1.5 is completely explicit: the operators with and uncountable which we construct are perturbations by a weighted backward shift on of a diagonal operator with unimodular diagonal coefficients. The construction here bears some similarities with a construction carried out in a different context in [11] in order to obtain frequently hypercyclic operators on certain Banach spaces. \par\par Let be a finite measure space where is a positive regular finite Borel measure, and let be a measurable transformation of . We recall here that is said to preserve the measure if for any , and that is said to be ergodic with respect to  if for any with and , there exists an such that , where denotes the iterate of . Equivalently, is ergodic with respect to  if and only if

 1NN∑n=1μ(φ−n(A)∩B)→μ(A)μ(B)%asN→+∞for every A,B∈F.

This leads to the notion of weakly mixing measure-preserving transformation of : is weakly mixing if

 1NN∑n=1|μ(φ−n(A)∩B)−μ(A)μ(B)|→0 as N→+∞for every A,B∈F.

It is well-know that is weakly mixing if and only if  is an ergodic transformation of endowed with the product measure . We refer the reader to [10], [31] or [39] for instance for more about ergodic theory of dynamical systems and various examples. \parOur interest in this paper lies in weakly mixing rigid dynamical systems:

###### Definition 1.6.  —

A measure-preserving transformation of is said to be rigid if there exists a sequence of integers such that for any , as .

If denotes the isometry on defined by for any , it is not difficult to see that is rigid with respect to the sequence if and only if  as for any . The function itself is said to be rigid with respect to  if . Rigid functions play a major role in the study of mildly mixing dynamical systems, as introduced by Furstenberg and Weiss in [13], and rigid weakly mixing systems are intensively studied, see for instance the works [16], [27], [17] or [37] as well as the references therein for some examples of results and methods. Let us just mention here the fact that weakly mixing rigid transformations of form a residual subset of the set of all measure-preserving transformations of for the weak topology [24]. A rigidity sequence is defined as follows: \par

###### Definition 1.7.  —

Let be a strictly increasing sequence of positive integers. We say that is a rigidity sequence if there exists a measure space and a measure-preserving transformation  of which is weakly mixing and rigid with respect to .

###### Remark 1.8.  —

In the literature one often defines rigidity sequences as sequences for which there exists an invertible measure-preserving transformation which is weakly mixing and rigid with respect to . In fact, these two definitions are equivalent since every rigid measure-preserving transformation  is invertible (in the measure-theoretic sense). An easy way to see it is to consider the induced isometry defined above. Since is invertible if and only if is so, it suffices to show that is invertible. By the decomposition theorem for contractions due to Sz.-Nagy, Foiaş [38], can be decomposed into a direct sum of a unitary operator and a weakly stable operator. Since in the weak operator topology (see Fact 3.2 below), the weakly stable part cannot be present and thus is a unitary operator and is invertible.

\par

Rigidity sequences are in a sense already characterized: is a rigidity sequence if and only if there exists a continuous probability measure  on the unit circle such that

 ∫T|λnk−1|dσ(λ)→0 as k→+∞

(see Section 3.1 for more details). Still, there is a lack of practical conditions which would enable us to check easily whether a given sequence is a rigidity sequence. It is the second aim of this paper to provide such conditions. Some of them can be initially found in the papers [2] and [3] which study Jamison sequences in the Banach space setting, and they turn out to be relevant for the study of rigidity. We show for instance that if tends to infinity as tends to infinity, is a rigidity sequence (see Example 3.4 and Proposition 3.5). If is any sequence such that divides for any , is a rigidity sequence (Propositions 3.8 and 3.9). We also give some examples involving the denominators of the partial quotients in the continuous fraction expansion of some irrational numbers (Examples 3.15 and 3.16), as well as an example of a rigidity sequence such that (Example 3.17). In the other direction, it is not difficult to show that if for some polynomial with for any , cannot be a rigidity sequence (Example 3.12), or that the sequence of prime numbers cannot be a rigidity sequence (Example 3.14). Other examples of non-rigidity sequences can be given (Example 3.13) when the sequences , , have suitable equirepartition properties. \par\par If is a bounded operator on a separable Banach space , it is sometimes possible to endow the space with a suitable probability measure , and to consider as a measurable dynamical system. This was first done in the seminal work [12] of Flytzanis, and the study was continued in the papers [5] and [6]. If is a separable complex Hilbert space which we denote by , admits a non-degenerate invariant Gaussian measure if and only if its eigenvectors associated to eigenvalues of modulus span a dense subspace of , and it is ergodic (or here, equivalently, weakly mixing) with respect to such a measure if and only if it has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues (see Section 2.1 for the definitions) – this condition very roughly means that has “plenty” of such eigenvectors, “plenty” being quantified by a continuous probability measure on the unit circle. \parIt comes as a natural question to describe rigidity sequences in the framework of linear dynamics, and it is not difficult to show that if is a rigidity sequence, there exists a bounded operator on which is weakly mixing and rigid with respect to  (see Section 4.1). Thus, every rigidity sequence can be realized in a linear Hilbertian measure-preserving dynamical system. However, the answer is not so simple when one considers topological and uniform rigidity, which are topological analogues of the (measurable) notion of rigidity. These notions were introduced by Glasner and Maon in the paper [15] for continuous dynamical systems on compact spaces: \par

###### Definition 1.9.  —

Let be a compact metric space, and let be a continuous self-map of . We say that is topologically rigid with respect to the sequence if as for any , and that is uniformly rigid with respect to  if

 supx∈Xd(φnk(x),x)→0 as k→+∞.
\par

It is easy to check, using the Lebesgue dominated convergence theorem, that a topologically or uniformly rigid dynamical system is rigid. Uniform rigidity is studied in [15], where in particular uniformly rigid and topologically weakly mixing transformations are constructed, see also [9], [25] and [21] for instance. Recall that is said to be topologically weakly mixing if for any non-empty open subsets of , there exists an integer such that and are both non-empty (topological weak mixing is the topological analogue of the notion of measurable weak mixing). Uniform rigidity sequences are defined in [21]: \par

###### Definition 1.10.  —

Let be a strictly increasing sequence of integers. We say that is a uniform rigidity sequence if there exists a compact dynamical system with a continuous self-map of , which is topologically weakly mixing and uniformly rigid with respect to .

\par

The question of characterizing uniform rigidity sequences is still open, as well as the question [21] whether there exists a compact dynamical system with continuous, which would be both weakly mixing with respect to a certain -invariant measure  on and uniformly rigid. \parWe investigate these two questions in the framework of linear dynamics. Of course we have to adapt the definition of uniform rigidity to this setting, as a Banach space is never compact. \par

###### Definition 1.11.  —

Let be complex separable Banach space, and let be a continuous transformation of . We say that is uniformly rigid with respect to  if for any bounded subset of ,

 supx∈A||φnk(x)−x||→0 as k→+∞.
\par

When is a bounded linear operator on , is uniformly rigid with respect to  if and only if  as . We prove the following theorems: \par

###### Theorem 1.12.  —

Let be an increasing sequence of integers with . The following assertions are equivalent:

• for any there exists a such that

 supk≥0|λnk−1|≤ε and |λnk−1|→0 as k→+∞;
• there exists a bounded linear operator  on a separable Banach space such that is uncountable and as for every ;

• there exists a bounded linear operator  on a separable Hilbert space such that admits a non-degenerate invariant Gaussian measure with respect to which is weakly mixing and as for every , i.e. is topologically rigid with respect to .

\par

We also have a characterization for uniform rigidity in the linear setting:

###### Theorem 1.13.  —

Let be an increasing sequence of integers. The following assertions are equivalent:

• there exists an uncountable subset of such that tends to uniformly on ;

• there exists a bounded linear operator  on a separable Banach space such that is uncountable and as ;

• there exists a bounded linear operator  on a separable Hilbert space such that admits a non-degenerate invariant Gaussian measure with respect to which is weakly mixing and as , i.e. is uniformly rigid with respect to .

\par

In particular we get a positive answer to a question of [21] in the context of linear dynamics: \par

###### Corollary 1.14.  —

Any sequence such that tends to infinity, or such that divides for each and is a uniform rigidity sequence for linear dynamical systems, and measure-theoretically weakly mixing uniformly rigid systems do exist in this setting.

\par\par\par

After this paper was submitted for publication, V. Bergelson, A. Del Junco, M. Lemańczyk and J. Rosenblatt sent us a preprint “Rigidity and non-recurrence along sequences” [8], in which they independently investigated for which sequences there exists a weakly mixing transformation which is rigid with respect to this sequence. A substantial part of the results of Section 3 of the present paper is also obtained in [8], often with different methods. We are very grateful to V. Bergelson, A. Del Junco, M. Lemańczyk and J. Rosenblatt for making their preprint available to us. \par\par\par Our aim in this section is to prove Theorem 1.5. Clearly, if is a Jamison sequence, it is automatically a Hilbertian Jamison sequence, and the difficulty lies in the converse direction: using Theorem 1.3, we start from a sequence such that for any there is a such that , and we have to construct out of this a bounded operator on a Hilbert space which is partially power-bounded with respect to  and which has uncountably many eigenvalues on the unit circle. We are going to prove a stronger theorem, giving a more precise description of the eigenvectors of the operator: \par

###### Theorem 2.1.  —

Let be an increasing sequence of integers with such that for any there exists a such that

 supk≥0|λnk−1|≤ε.

Let be any positive number. There exists a bounded linear operator  on the Hilbert space such that has perfectly spanning unimodular eigenvectors and

 supk≥0||Tnk||≤1+δ.

In particular the unimodular point spectrum of is uncountable.

\par

Before embarking on the proof, we need to define precisely the notion of perfectly spanning unimodular eigenvectors and explain its relevance here. \par\par Let be a complex separable infinite-dimensional Hilbert space. \par

###### Definition 2.2.  —

We say that a bounded linear operator on has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues if there exists a continuous probability measure  on the unit circle such that for any Borel subset of with , we have .

\par

When has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues, there exists a Gaussian probability measure  on such that: \par is -invariant; \par is non-degenerate, i.e. for any non-empty open subset of ; \par is weakly mixing with respect to . \parSee [6] for extensions to the Banach space setting, and the book [7, Ch. 5]. In the Hilbert space case, the converse of the assertion above is also true: if defines a weakly mixing measure-preserving transformation with respect to a non-degenerate Gaussian measure, has perfectly spanning unimodular eigenvectors. \parA way to check this spanning property of the eigenvectors is to use the following criterion, which was proved in [18, Th. 4.2]: \par

###### Theorem 2.3.  —

Let be a complex separable infinite-dimensional Banach space, and let be a bounded operator on . Suppose that there exists a sequence of vectors of having the following properties:

• for each , is an eigenvector of associated to an eigenvalue  of where and the ’s are all distinct;

• is dense in ;

• for any and any , there exists an such that .

Then has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues. \par

\par\par

We are first going to define the operator , and show that it is bounded. We will then describe the unimodular eigenvectors of , and show that satisfies the assumptions of Theorem 2.3. \par Construction of the operator . Let denote the canonical basis of the space of complex square summable sequences. We denote by the space . The construction depends on two sequences and which will be suitably chosen further on in the proof: is a sequence of unimodular complex numbers which are all distinct, and is a sequence of positive weights. \parLet be a function having the following two properties: \par for any , ; \par for any , the set is infinite (i.e. takes every value infinitely often). \parLet be the diagonal operator on defined by for , and let be the weighted backward shift defined by and for , where the weights , , are defined by

 α1=ω1|λ2−λj(2)|

and

 αn=ωn∣∣∣λn+1−λj(n+1)λn−λj(n)∣∣∣for any n≥2.

This definition of makes sense because , so that . The operators  and being thus defined, we set . \par Boundedness of the operator . The first thing to do is to prove that is indeed a bounded operator on , provided some conditions on the ’s and ’s are imposed. The diagonal operator  being obviously bounded, we have to figure out conditions for to be bounded. If is fixed, we have provided

 ω1|λ2−λj(2)|≤γandωn−1∣∣∣λn−λj(n)λn−1−λj(n−1)∣∣∣≤γfor any n≥3.

If the weights are arbitrary, the ’s can be chosen in such a way that these conditions are satisfied: \par is arbitrary, we take for instance (we could start here from any ); \par we have : take such that with ; \par take arbitrary; \par : take so close to , , that

 |λ3−λj(3)|≤γω2|λ2−λj(2)|
\par

take arbitrary, etc. \parThus provided is so close to for every that

 |λn−λj(n)|≤γωn−1|λn−1−λj(n−1)|.

No condition on the ’s needs to be imposed there. \par Unimodular eigenvectors of the operator . The algebraic equation with is equivalent to the equations , i.e. for any , i.e.

 xk=(λ−λk−1)…(λ−λ1)αk−1…α1x1.

Hence for any , the eigenspace is -dimensional and , where

 u(n)=e1+n∑k=2(λn−λk−1)…(λn−λ1)αk−1…α1ek.

Our aim is now to show the following lemma:

###### Lemma 2.4.  —

By choosing in a suitable way the coefficients and , it is possible to ensure that for any ,

 ||u(n)−u(j(n))||≤2−n

(the sequence could be replaced by any sequence going to zero with ).

\par
###### Proof of Lemma 2.4.

We have:

 u(n)−u(j(n)) = j(n)∑k=2((λn−λk−1)…(λn−λ1)αk−1…α1−(λj(n)−λk−1)…(λj(n)−λ1)αk−1…α1)ek + n∑k=j(n)+1(λn−λk−1)…(λn−λ1)αk−1…α1ek:=v(n)+w(n).

We denote the first sum by and the second one by . If is any positive number, we can ensure that by choosing such that is sufficiently small, because the quantities for do not depend on . Let us now estimate

 ||w(n)||2 = n∑k=j(n)+1∣∣∣(λn−λk−1)…(λn−λ1)αk−1…α1∣∣∣2 = n∑k=j(n)+11ω2k−1…ω21⋅∣∣∣(λn−λk−1)…(λn−λ1)λk−λj(k)∣∣∣2

since

 αk−1…α1=ωk−1…ω1|λk−λj(k)|.

We estimate now each term in this sum. We can suppose that for any and (this is no restriction), so since . Thus for ,

 1ω2k−1…ω21⋅∣∣∣(λn−λk−1)…(λn−λ1)λk−λj(k)∣∣∣2≤1ω2k−1…ω21⋅∣∣∣λn−λj(n)λk−λj(k)∣∣∣2⋅

If , the term on the right-hand side can be made arbitrarily small provided that we choose in such a way that is very small with respect to the quantities , . However for , we only get the bound which has to be made small if we want to be small. So we have to impose a condition on the weights : we take so large with respect to that is extremely small. \parAll the conditions on the ’s and the ’s needed until now can indeed be satisfied simultaneously: at stage of the construction, we take very large. After this we take extremely close to . Thus we can ensure that , hence that . Taking gives our statement. ∎

\par

Thanks to Lemma 2.4, it is easy to see that satisfies the assumptions of Theorem 2.3: \par

###### Proposition 2.5.  —

The operator  satisfies the assumptions of Theorem 2.3. Hence it has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues, and in particular its unimodular point spectrum is uncountable.

\par
###### Proof of Proposition 2.5.

It suffices to show that the sequence satisfies properties (i), (ii) and (iii). That (i) is satisfied is clear, since the vectors are eigenvectors of associated to the eigenvalues which are all distinct. Since for each the vector belongs to the span of the first basis vectors and , the linear span of the vectors , , contains all finitely supported vectors of , and thus (ii) holds true. It remains to prove (iii). As the function takes every value in infinitely often, it follows from Lemma 2.4 that for any there exists a strictly increasing sequence of integers such that

 ||u(p(n)s)−u(n)|| tends to 0 as s tends % to +∞.

Hence (iii) is true. ∎

\par

In order to conclude the proof of Theorem 1.5, it remains to show that is partially power-bounded with respect to , with . This is the most difficult part of the proof, which uses the assumption that is not a Jamison sequence, and it is the object of the next section. \par\par In order to estimate the norms , we will show that provided the ’s and ’s are suitably chosen, for every . Since , this will prove that for every . \par An expression of . We first have to compute for and . For , let be the coefficient in row and column of the matrix representation of . If , (all coefficients below the diagonal are zero), and if , (all coefficients which are not in one of the first upper diagonals of the matrix are zero). We have for any . \par\par

###### Lemma 2.6.  —

For any such that ,

 t(n)k,l=αl−1αl−2…αk∑jk+…+jl=n−(l−k)λjkk…λjll.
\par
###### Proof.

The proof is done by induction on . \par : in this case , and the formula above gives , which is true. \par Suppose that the formulas above are true for any . Let and be such that (in particular ). We have

 t(n+1)k,l=t(n)k,l−1t(1)l−1,l+t(n)k,lt(1)l,l=αl−1t(n)k,l−1+λlt(n)k,l.

If , we can apply the induction assumption to the two quantities and , and we get

 t(n+1)k,l = αl−1αl−2…αk∑jk+…+jl−1=n−(l−1−k)λjkk…λjl−1l−1 + αl−1αl−2…αk∑jk+…+jl=n−(l−k)λjkk…λjl−1l−1λjl+1l = ∑jk+…+jl−1+jl=n+1−(l−k),jl=0λjkk…λjl−1l−1λjll + ∑jk+…+jl−1+jl=n+1−(l−k),jl≥1λjkk…λjl−1l−1λjll = ∑jk+…+jl−1+jl=n+1−(l−k)λjkk…λjl−1l−1λjll

and the formula is proved for . It remains to treat the cases where and where . If , we have By the induction assumption

 t(n)k,k+1=αk∑jk+jk+1=n−1λjkkλjk+1k+1=αkλnk+1−λnkλk+1−λk

so that

 t(n+1)k,k+1=αk(λnk+λk+1λnk+1−λnkλk+1−λk)=αkλn+1k+1−λn+1kλk+1−λk=αk∑jk+jk+1=nλjkkλjk+1k+1

which is the formula we were looking for. Lastly, when , . By the induction assumption , thus and the formula is proved in this case too. ∎

\par

A first estimate on the norms . For , let us estimate : we have

so that by the Cauchy-Schwarz inequality

 ||(Tn−Dn)x||2≤||x||2∑l≥2∣∣ ∣∣∣∣ ∣∣l−1∑k=max(1,l−n)t(n)k,lek∣∣ ∣∣∣∣ ∣∣2≤||x||2∑l≥2l−1∑k=max(1,l−n)|t(n)k,l|2.

We thus have to estimate for each and the quantities

 l−1∑k=max(1,l−np)|t(np)k,l|2.

For , , let

 s(n)k,l=∑jk+…+jl=n−(l−k)λjkk…λjll.

We have

 t(n)k,l=αl−1…αks(n)k,l=ωl−1…ωk|λl−λj(l)||λk−λj(k)|s(n)k,l

so that we have to estimate

 l−1∑k=max(1,l−n)ω2l−1…ω2k|λl−λj(l)|2|λk−λj(k)|2|s(n)k,l|2.

We are going to show that the following property holds true: \par