aa r X i v : . [ m a t h . F A ] A ug COMMON FREQUENT HYPERCYCLICITY
S. CHARPENTIER, R. ERNST, M. MESTIRI, A. MOUZE
Abstract.
We provide with criteria for a family of sequences of operators to share a fre-quently universal vector. These criteria are variants of the classical Frequent HypercyclicityCriterion and of a recent criterion due to Grivaux, Matheron and Menet where periodicpoints play the central role. As an application, we obtain for any operator T in a specificclass of operators acting on a separable Banach space, a necessary and sufficient conditionon a subset Λ of the complex plane for the family { λT : λ ∈ Λ } to have a common fre-quently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclicweighted shifts which do not possess common frequent hypercyclic vectors. We also providewith criteria for families of the recently introduced operators of C -type to share a com-mon frequently hypercyclic vector. Further, we prove that the same problem of common α -frequent hypercyclicity may be vacuous, where the notion of α -frequent hypercyclicity ex-tends that of frequent hypercyclicity replacing the natural density by more general weighteddensities. Finally, it is already known that any operator satisfying the classical FrequentUniversality Criterion is α -frequently universal for any sequence α satisfying a suitable con-dition. We complement this result by showing that for any such operator, there exists avector x which is α -frequently universal for T , with respect to all such α . Introduction
For two separable Fréchet spaces X and Y , let us denote by L ( X, Y ) the set of all con-tinuous operators from X to Y . If X = Y , we simply write L ( X ) = L ( X, Y ) . A sequence T = ( T n ) n ∈ N ⊂ L ( X, Y ) (where N := { , , , . . . } ) is said to be universal provided thereexists a vector x ∈ X such that for any non-empty open subset U of Y , the set N ( x, U, T ) := { n ∈ N : T n x ∈ U } is infinite. The vector x is also called universal and the set of all universal vectors for T isdenoted by U ( T ) . A single operator T ∈ L ( X ) is called hypercyclic if the sequence ( T n ) n ∈ N ofits iterates is universal. In this case, we write N ( x, U, T ) = N ( x, U, T ) and U ( T ) = HC ( T ) .In 2006, Bayart and Grivaux [4] introduced the important notion of frequently hypercyclicoperator . An operator T ∈ L ( X ) is said to be frequently hypercyclic if there exists x ∈ X suchthat for any non-empty open subset U of X , the lower density d ( N ( x, U, T )) of N ( x, U, T ) is positive, where for any E ⊂ N , d ( E ) := lim inf n card ([0 , n ] ∩ E ) n + 1 > . Such a vector x is a frequently hypercyclic vector for T and the set of such vectors is denotedby F HC ( T ) . The notion of frequent universality for a sequence T of operators in L ( X, Y ) can obviously be defined (see, for e.g., [11]). The set of frequently universal vectors for T willbe denoted by F U ( T ) . For a rich source of information about Linear Dynamics, we refer tothe monographs [7, 26].A problem which has been extensively studied during the last decades is that of commonhypercyclicity . For a given family ( T λ ) λ ∈ Λ of hypercyclic operators in L ( X ) , it asks when the Mathematics Subject Classification.
Key words and phrases.
Frequently hypercyclic operator, weighted shift operator.The first, second and fourth authors are supported by the grant ANR-17-CE40-0021 of the French NationalResearch Agency ANR (project Front). The third author is supported by a grant of F.R.S.-FNRS. set of common hypercyclic vectors , T λ ∈ Λ HC ( T λ ) , is empty and when it is not. [7, Chapter7] and [26, Chapter 11] are entirely devoted to this topic. On the one hand, since HC ( T ) is a dense G δ subset of X whenever it is non-empty, the Baire Category Theorem triviallyensures that T λ ∈ Λ HC ( T λ ) is non-empty whenever Λ is countable. On the other hand,it is not difficult to exhibit families of hypercyclic operators with no common hypercyclicvectors (for example the family of all hypercyclic operators on a given space X ). Thefirst positive important result in this direction was given by Abakumov and Gordon [1]who showed that T λ> HC ( λB ) = ∅ , where B is the backward shift on ℓ ( N ) defined by B ( x , x , x , . . . ) = ( x , x , x , . . . ) . Later on, Costakis and Sambarino [18] provided with thefirst criterion of common hypercyclicity that they applied to show the residuality of the set ofcommon hypercyclic vectors for multiples of the backward shift or differential operators, andfor uncountable families of translation operators or specific weighted shifts. Constructions orthe approach used by Costakis and Sambarino, based on the Baire Category Theorem, weredeveloped by many authors to produce new criteria or prove common hypercyclicity for otheruncountable families of classical operators, such as adjoint of multipliers, or composition andconvolution operators (see, for e.g., [2, 5, 6, 13, 16, 24]). A second approach to the problem,more algebraic, produced some of the most striking results. León and Müller proved thatfor any T ∈ L ( X ) and any λ ∈ C , | λ | = 1 , HC ( T ) = HC ( λT ) . Their idea, which exploitsthe group structure of the torus T = { z ∈ C : | z | = 1 } , was extended by several authors tofamilies of operators forming groups or semigroups, and then combine with the first approachto produce some new and strong results (for e.g., [3, 9, 15, 32, 34]). We should say that thenon-existence of common universal vectors has also been studied (see, for e.g., [3, 7, 19, 26]).In comparison, common frequent hypercyclicity has been considered in only a very fewamount of papers. Probably, it is partly because the Baire Category approach drasticallyfails for this notion: by [8, Corollary 19], the set F HC ( T ) is always meager (i.e., containedin the complement of a residual set). Moreover, for any T ∈ L ( X ) , it turns out that theset T λ ∈ Λ F HC ( λT ) is empty, as soon as Λ ⊂ (0 , + ∞ ) is uncountable ([3, Proposition 6.4]).However, the algebraic approach to common hypercyclicity perfectly fits to frequent hyper-cyclicity. For example, Bayart and Matheron proved that F HC ( λT ) = F HC ( T ) for any λ ∈ T , obtaining a frequent version of León-Müller’s result. This approach has been pursuedfurther in [3] (see also [15]) and led to several nice results of common frequent hypercyclicityfor families of operators forming strongly continuous groups or semigroups (translation op-erators on H ( C d ) , composition operators induced by non-constant Heisenberg translationson the Hardy space of the Siegel half-space, etc...). Moreover, in specific classes of opera-tors, hypercyclic basically means frequently hypercyclic in a strong sense. For example, if Λ hyp denotes the set of all hyperbolic automorphisms of the unit disc D having the sameboundary attractive point, then the same argument as in [7, Example 7.3] gives that thereexists φ ∈ Λ hyp such that for any φ ∈ Λ hyp , F HC ( C φ ) ⊂ F HC ( C φ ) , where C φ denotesthe composition operator with symbol φ on the Hardy space H of D . Combined with thealgebraic approach, this yields T φ ∈ Λ F HC ( C φ ) = ∅ where Λ stands for the set of all auto-morphisms having a common boundary attractive point. All in all, except when action bystrongly continuous groups or semigroups is involved, so far no criteria for common frequenthypercyclicity are known. In particular, we do not know under which non-trivial conditionson Λ ⊂ (0 , + ∞ ) and T ∈ L ( X ) the set T λ ∈ Λ F HC ( λT ) may be non-empty.In this paper we aim to contribute in filling these gaps. Our first result is a criterion ofcommon frequent universality (Theorem 2.3) which is a natural strengthening of the FrequentUniversality Criterion given in [12] (and of the classical Frequent Hypercyclicity Criterion[4]). As an application, we get necessary and/or sufficient conditions on a subset Λ of C forthe set T λ ∈ Λ F HC ( λT ) to be non-empty, when X is a Banach space and T ∈ L ( X ) . Forexample, we will get the following: OMMON FREQUENT HYPERCYCLICITY 3
Theorem.
Let B be the backward shift on ℓ ( N ) and let Λ ⊂ C . The set T λ ∈ C F HC ( λT ) is non-empty if and only if the set {| λ | : λ ∈ Λ } is a countable relatively compact subset of (1 , + ∞ ) . This theorem is obtained for more general classes of (unilateral) weighted shifts on ℓ ( N ) .For any operator T ∈ L ( X ) , sufficient or necessary conditions on Λ are given, involving thespectral or the local spectral radius of T . In full generality, our sufficient condition exactlycoincides with the assumption of a criterion of common hypercyclicity given by Bayart andMatheron [6, Proposition 4.2]. Our general criterion of common frequent universality is alsoapplied to countable families of weighted shifts, differential operators or adjoint of multipliers(which may not be multiples of a single operator). In passing, we deduce a simple way toproduce two frequently hypercyclic weighted shifts without common frequently hypercyclicvectors.Recently, Grivaux, Matheron and Menet provided with a new frequent hypercyclicity cri-terion, based on the periodic points of the operator [25]. They prove that this criterion istheoretically better than the classical Frequent Hypercyclicity Criterion since any operatorsatisfying the assumptions of the latter automatically satisfies that of the new one. In prac-tice, the classical criterion turns out to be much simpler to apply to most of the explicit andusual operators. However, Menet introduced a new class of operators, the so-called operatorsof C -type [27], conceived as a very rich source of counter-examples to difficult problems (suchas the exhibition of a chaotic operator on ℓ p which is not frequently hypercyclic [27], see also[25]), to which their new criterion for frequent hypercyclicity is very well adapted. In thepresent paper, based on this criterion, we establish another general criterion for commonfrequent hypercyclicity, involving the periodic points of the family of operators. Once again,we show how this can be easily applied to classes of operators of C -type.Furthermore, Ernst and Mouze recently proved [20, 21] that any operator satisfying theusual Frequent Universality Criterion in fact enjoys a stronger form of frequent universality.Let α = ( α k ) k ≥ be a sequence of non-negative real numbers with P k ≥ α k = + ∞ . In [23],Freedman and Sember show that if a matrix ( α n,k ) n,k ≥ is given by α n,k = (cid:26) α k / ( P nj =1 α j ) for ≤ k ≤ n, otherwise.then the function d α : P ( N ) → [0 , ( P ( N ) denotes the set of all subsets of N ) defined for E ⊂ N by d α ( E ) = lim inf n X k ≥ α n,k E ( k ) ! is a generalized lower density (see [23] for the abstract definition of a (generalized lower)density). We call d α ( E ) the lower α -density of E . The usual lower density encounteredabove corresponds to the constant sequence (1 , , , . . . ) . Moreover, if α . β (meaning α k /β k is eventually decreasing to ), then d β ( E ) ≤ d α ( E ) , E ⊂ N ([20, Lemma 2.8]). The order . thus allows to define (ordered) scales of generalized lower densities. We refer to [20, 21]for examples of sequences α defined by usual functions and well-ordered with respect to . .It then appears natural to define α -frequent universality as the usual frequent universality,replacing the sequence (1 , , , . . . ) by any α as above. One of the main results of [20, 21]is that any operator T ∈ L ( X ) which satisfies the Frequent Universality Criterion is d α -frequently universal whenever there exists s ≥ such that α . (exp( k/ (log ( s ) ( k )))) k ≥ where log ( s ) = log ◦ log ◦ . . . ◦ log , log appearing s times. Moreover, they prove that no operatorcan be α -frequent hypercyclic for α k = e k . In view of the topic of the paper, two natural S. CHARPENTIER, R. ERNST, M. MESTIRI, A. MOUZE questions arise. For T ∈ L ( X ) we denote by F HC α ( T ) the set of all α -frequent hypercyclicvectors for T . Questions.
Let A denote the set of sequences α such that α . d s for some s ≥ and let T ∈ L ( X ) .1) Let Λ ∈ (0 , + ∞ ) and B ⊂ A be non-trivial. Do we have T ( λ,β ) ∈ Λ × B F HC β ( λT ) = ∅ ?
2) If T satisfies the Frequent Hypercyclicity Criterion, do we have T α ∈ A F HC α ( T ) = ∅ ? We will give a positive answer to the second question (Proposition 4.6) and show thatthe first one has a strongly negative answer if Λ is any non-trivial subset of (0 , + ∞ ) and B is reduced to a single generalized density which grows faster than ( e k ε ) k ≥ for some ε > (Proposition 4.2). We should mention that, by [20, Lemma 2.10], F HC β ( T ) = F HC ( T ) whenever β has a growth at most polynomial (i.e., β . ( k r ) k ≥ for some r ≥ − ). Combinedwith our first common frequent hypercyclicity criterion, this thus gives a positive answer to(1) for some non-trivial Λ and the set B of sequences with at most polynomial growth.We should conclude by mentioning that the problem of common hypercyclicity has beenconsidered for the upper (or U -)frequent hypercyclicity. This intermediate notion betweenhypercyclicity and frequent hypercyclicity was introduced by Shkarin [33]. A sequence T ⊂L ( X, Y ) is said to be U -frequently universal if for some x ∈ X and any non-empty openset U in Y , d ( N ( x, U, T )) is positive. By definition, d ( E ) = 1 − d ( N \ E ) is the upperdensity of E ⊂ N . In some sense, U -frequent hypercyclicity is closer to hypercyclicity thanto frequent hypercyclicity. For example, Bayart and Ruzsa proved that the set U F HC ( T ) ofall U -frequently hypercyclic vectors for T is residual whenever it is non-empty [8, Proposition21]. Common U -frequent hypercyclicity has been rather well-studied and criteria have beengiven. We refer to [28, 29] and the references therein for an up-to-date and complete overviewon the subject. In the sequel, we shall (almost) not come back to this notion.The paper is organized as follows. Section 2 is devoted to our first general criteria ofcommon frequent universality and their applications. In Section 3, we focus on the statementof our second criterion for common frequent hyperyclicity involving periodic points. Wefinally give the answers to the last two questions in Section 4.2. Common frequent universality for countable families of operators
A general criterion.
For the proof of the main result of this section, we will makeuse of the following refinement of [7, Lemma 6.19] and of ideas developed in [8].
Lemma 2.1.
For every
K > and every countable family ( N p ( i )) p , i ∈ N , of increasingsequences of positive integers, there exists a countable family ( E p ( i )) p , i ∈ N , of sequences ofsets E p ( i ) ⊂ N with positive lower density, such that for every ( p, i ) , ( q, j ) ∈ N ,(1) min( E p ( i )) ≥ N p ( i ) ;(2) For every n ∈ E p ( i ) , m ∈ E q ( j ) , [ n = m = ⇒ | n − m | ≥ N p ( i ) + N q ( j ) ≥ max( N p ( i ) , N q ( j ))] ;(3) If ( p, i ) = ( q, j ) , then for every n ∈ E p ( i ) , m ∈ E q ( j ) , [ n > m = ⇒ n ≥ Km ] ;Proof. Let
K > and for every i ∈ N , let ( N p ( i )) p be an increasing sequences of positiveintegers. Moreover, for every i ∈ N , let us denote by ( A p ( i )) p a sequence of syndetic setsforming a partition of N . Let also < ε < and a > be such that − ε ε a > K. For every u ∈ N , we pose: I a,εu = [(1 − ε ) a u ; (1 + ε ) a u ] OMMON FREQUENT HYPERCYCLICITY 5 and we set: E p ( i ) = ∪ u ∈ A p ( i ) ( I a,εu ∩ (2 N p ( i ) N )) . Remark that by definition, (2) is satisfied when ( p, i ) = ( q, j ) . Remark also that for every u ∈ A p ( i ) , we have the following equivalence: I a,εu + [ − N p ( i ); N p ( i )] ⊆ I a, εu ⇔ N p ( i ) ≤ εa u . Thus, it suffices to remove a finite number of elements in each A p ( i ) to ensure that both con-ditions above are satisfied. Moreover, such a modification of the sets A p ( i ) has no influenceon rest the proof.In the same spirit, one may check that for every u > v , I a, εu ∩ I a, εv = ∅ ⇔ < − ε ε a u − v . Moreover, the choice we made on a and ε gives < K < − ε ε a < − ε ε a u − v . Therefore, for every u > v , I a, εu ∩ I a, εv = ∅ . This last relation and the previous one provethat (2) is satisfied.To check that (3) is satisfied, remark first that our assumptions on a and ε implies that − ε ε a > K . Then, for every n > m with n ∈ E p ( i ) and m ∈ E q ( j ) , there exists u ∈ A p ( i ) and v ∈ A q ( j ) with u > v so that: ( (1 − ε ) a u ≤ n ≤ (1 + ε ) a u (1 − ε ) a v ≤ m ≤ (1 + ε ) a v Thus, we have: Km ≤ K (1 + ε ) a v < (1 − ε ) a v +1 ≤ (1 − ε ) a u ≤ n. This proves (3)Condition (1) is easy to obtain, up to removing a finite number of elements from each set E p ( i ) which does not modify the other conditions.Finally, it remains to prove that each set E p ( i ) has positive lower density. Let p, i ∈ N and ( n k ) k ∈ N be an enumeration of the set A p ( i ) and M be the maximal size of a gap in A p ( i ) .Then, d ( E p ( i )) = lim inf k →∞ E p ( i ) ∩ [0; (1 + ε ) a n k ])(1 − ε ) a n k +1 ≥ lim inf k →∞ (cid:18) εa n k N p ( i ) − (cid:19) a n k +1 ≥ lim inf k →∞ (cid:18) εa n k N p ( i ) − (cid:19) a n k + M = εN p ( i ) a M > This ends the proof of the lemma. (cid:3)
We recall the definition of uniform unconditional convergence.
Definition 2.2.
Let Λ be a set. We say that the series P x λ,n , λ ∈ Λ in X convergesunconditionally uniformly for λ ∈ Λ if, for every ε > , there is some N ∈ N such that forany finite set F ⊂ { N, N + 1 , . . . } , one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ F x λ,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε S. CHARPENTIER, R. ERNST, M. MESTIRI, A. MOUZE for every λ ∈ Λ .Our general common frequent universality criterion for countable families of operatorsstates as follows. Theorem 2.3.
Let X be an F -space, Y a separable F -space and ( T i,n ) n ∈ N , i ∈ N , be countablymany sequences of continuous linear operators from X to Y . We assume that there exists adense subset Y of Y , mappings S i,n : Y → X , i, n ∈ N , and a real number c > such thatfor every y ∈ Y ,(1) The series P mn =0 T i,m ( S i,m − n ( y )) converges unconditionally, uniformly for m ∈ N and i ∈ N ;(2) The series P n ≥ T i,m ( S i,m + n ( y )) converges unconditionally, uniformly for m ∈ N and i ∈ N ;(3) The series P n ≥ ( c − m T i,m ( S j,m + n ( y )) converges unconditionally, uniformly for m ∈ N and i = j ∈ N ;(4) The series P c − c m ≤ n ≤ m T i,m ( S j,m − n ( y )) converges unconditionally, uniformly for m ∈ N and i = j ∈ N ;(5) The series P n ≥ S i,n ( y ) converges unconditionally, uniformly for i ∈ N ;(6) The sequence ( T i,n ( S i,n ( y ))) converges to x , uniformly for every i ∈ N .Then there exists a vector x ∈ X frequently universal for every ( T i,n ) n , i ∈ N . One can easily check that each ( T i,n ) n , i ∈ N satisfies (1), (2), (5) and (6) if and only if itsatisfies the Frequent Universality Criterion given in [12]. Proof.
Since Y is separable, we can assume that Y = { y , y , . . . } . Let ( ε p ) p ∈ N be a decreasingsequence of positive real numbers such that P p ε p < and pε p → as p → ∞ . We alsofix an increasing sequence ( J p ) p such that P i ≥ J p ε i < ε p . The assumptions of the theoremimply the existence of a sequence ( N p ( i )) i,p ∈ N such that for every i, p ∈ N , every finite set F ⊂ { N p ( i ) , N p ( i ) + 1 , . . . } , every m ∈ N , every q ∈ { , . . . , p } , every k ∈ N and every n ≥ N p ( i ) ,(i) k X n ∈ Fn Note that x is even unconditionally convergent. Our goal is now to prove that x is a frequentlyuniversal vector for each sequence ( T i,n ) n , i ∈ N . We fix j ∈ N . Let ( r q ) q be a sequenceof positive real numbers with r q → as q → ∞ , to be chosen later. Since the sets E p ( i ) , i, p ∈ N , have positive lower density, it is sufficient to prove that(2.1) k T j,m ( x ) − y q k < r q for every j ∈ N , q ∈ N and every m ∈ E q ( j ) . Using that E p ( i ) ∩ E q ( j ) = ∅ if ( i, p ) = ( j, q ) and that x is unconditionally convergent in X , if m ∈ E q ( j ) then we can decompose T j,m ( x ) as follows: T j,m ( x ) = T j,m ( S j,m ( y q )) + A m z }| {X p ∈ N X n ∈ E p ( j ) n = m T j,m ( S j,n ( y p )) + B m z }| {X i ∈ N i = j X p ∈ N X n ∈ E p ( i ) n = m T j,m ( S i,n ( y p )) . First, since m ≥ N q ( j ) for any m ∈ E q ( j ) , (viii) gives(2.2) k T j,m ( S j,m ( y q )) − y q k < ε q . We next estimate A m : k A m k ≤ X p ∈ N k X n ∈ E p ( j ) n Let ( T i ) i ∈ N be countably many bounded linear operators on X . We assumethat there exists a dense subset X of X , mappings S i : X → X , i ∈ N , and a real number c > such that for every x ∈ X , OMMON FREQUENT HYPERCYCLICITY 9 (1) The series P n ≥ T ni ( x ) and P n ≥ S ni ( x ) converge unconditionally, uniformly for i ∈ N ;(2) The sequence T i S i ( x ) = x for every i ∈ N ;(3) The series P n ≥ ( c − m T mi ( S m + nj ( x )) converges unconditionally, uniformly for m ∈ N and i = j ∈ N ;(4) The series P c − c m ≤ n ≤ m T mi ( S m − nj ( x )) converges unconditionally, uniformly for m ∈ N and i = j ∈ N .Then there exists a vector x ∈ X frequently hypercyclic for every T i , i ∈ N . In the previous statement, (1) and (2) exactly say that each T i satisfies the FrequentHypercyclicity Criterion. Note that the second part of (1) is a consequence of (3) by taking m = 0 .These two results apply to many situations, that we describe below.2.2. Application to multiples of a single operator. Let fix a continuous linear operator T on X . Given X a dense subset of X and S : X → X such that T S ( x ) = x for x ∈ X ,we denote by a T ( X , S ) = inf { λ : X S n λ n ( x ) converges unconditionally for all x ∈ X } = inf { λ : ( λ − n S n ( x )) n ∈ N is bounded for all x ∈ X } . and b T ( X , S ) = sup { λ : X ( λT ) n ( x ) converges unconditionally for all x ∈ X } = sup { λ : (( λT ) n ( x )) n ∈ N is bounded for all x ∈ X } . One easily checks that a T ( X , S ) = sup x ∈ X lim sup n k S n ( x ) k /n and b T ( X , S ) = inf x ∈ X n k T n ( x ) k /n . In particular, if X is a Banach space and r ( T ) denotes the spectral radius of T ,(2.13) a T ( X , S ) ≥ n k T n k /n = 1lim n k T n k /n = 1 r ( T ) ≥ k T k , where the equality follows from the spectral radius formula. Also note that b T ( X , S ) may beinfinite, for e.g., if X = S n ≥ ker T n is dense in X . This is for example the case if T is anyweighted backward shift acting on the Fréchet space X with an unconditional basis. Evenmore specifically, if T is the unweighted backward shift B on ℓ ( N ) , then S can be taken asthe unweighted forward shift F and we have equalities in (2.13) with a B ( X , F ) = 1 / k B k = 1 (see Paragraph 2.3 for a focus on weighted shifts).It is not difficult to check (see Lemma 2.7 below) that if a T ( X , S ) < λ < b T ( X , S ) then λT satisfies the Frequent Hypercyclicity Criterion ([7, Theorem 6.18]). The followingcriterion of common hypercyclicity, due to Bayart and Matheron ([6, Proposition 4.2]), canbe rephrased as follows. Theorem 2.5 (Bayart-Matheron) . Let X be a separable Fréchet space and let T : X → X bea continuous linear operator. Assume that there exist X ⊂ S n ∈ N ker( T n ) and S : X → X such that X is dense in X and T S ( x ) = x for all x ∈ X . Then T λ>a T ( X ,S ) HC ( λT ) is adense G δ subset of X . It is known that for any continuous operator T on X , T λ ∈ Λ F HC ( λT ) = ∅ whenever Λ is anuncountable subset of (0 , + ∞ ) (of course, even if λT is frequently hypercyclic for any λ ∈ Λ ),see [26, Exercise 9.2.7]. Under the same (in fact, a bit weaker) assumptions as in Bayart-Matheron’s criterion, we have the following countably common frequent hypercyclicity. Theorem 2.6. Let X be a separable Banach space and let T : X → X be a continu-ous linear operator. Assume that there exists a dense subset X of X and S : X → X such that T S ( x ) = x for all x ∈ X . If Λ is a countable relatively compact subset of ( a T ( X , S ) , b T ( X , S )) , then T λ ∈ Λ F HC ( λT ) = ∅ . The proof of this theorem is based on the following easy lemma, where it is assumed that E ⊂ ( a, b ) with b < a means E = ∅ . Lemma 2.7. With the notations of Theorem 2.6, let E be a relatively compact subset of ( a T ( X , S ) , b T ( X , S )) . Then there exists c > such that for any x ∈ X ,(i) The series P n ≥ ( λT ) n ( x ) converges unconditionally, uniformly for λ ∈ E ;(ii) The series P n ≥ (cid:0) Sλ (cid:1) n ( x ) converges unconditionally, uniformly for λ ∈ E ;(iii) The series P n ≥ ( c − m ( λµ ) m ( Sµ ) n ( x ) converges unconditionally, uniformly for m ∈ N and λ, µ ∈ E .(iv) The series P n ≥ c − c m ( λµ ) m − n ( λT ) n ( x ) converges unconditionally, uniformly for m ∈ N and λ, µ ∈ E .Proof. For notational simplicity, we shall denote a = a T ( X , S ) and b = b T ( X , S ) . We onlyprove (ii) and (iv). (i) and (iii) are respectively proved in the same way. To get (ii), let a < d < inf( E ) . Then, it is enough to write, for λ ∈ E , ( Sλ ) n ( x ) = ( dλ ) n ( Sd ) n ( x ) , and use that dλ ≤ d inf( E ) < and that ( Sd ) n ( x ) is bounded for any x ∈ X by some constant independentof λ ∈ E and n ∈ N .To prove (iv), let us now fix d ∈ ( a, b ) such that sup( E ) < d < b . Then P ( dT ) n ( x ) isconvergent in X and the sequence (( dT ) n ( x )) n is bounded for any x ∈ X by some constant M independent of m and λ, µ ∈ E . Then, for any c > and any λ, µ ∈ E , we have, writing n = c − c m + s + k with k ∈ N and ≤ s < which does depend on m and c but not on n , (cid:18) λµ (cid:19) m − n k ( λT ) n ( x ) k = λ m µ m − ( c − c m + s + k ) d c − c m + s + k k ( dT ) n ( x ) k≤ M (cid:18) λ ( µd c − ) /c (cid:19) m (cid:18) sup( E ) d (cid:19) k + s . Now, one can observe that λ ( µd c − ) /c is less than uniformly for λ, µ ∈ E whenever ( sup( E ) d ) c < ab , which in turn holds true whenever c is large enough. (cid:3) Let us now finish the proof of Theorem 2.6. Proof of Theorem 2.6. It is enough to check that the sequences (( λT ) n ) n and (( S/λ ) n ) n , λ ∈ Λ , satisfy the assumptions (1)–(6) of Theorem 2.3. (6) is trivial, while (1), (2) and (5)are direct consequences of (i) and (ii) of Lemma 2.7. Now, (3) and (4) follow from (iii) and(iv) of Lemma 2.7, after observing that for any λ = µ ∈ Λ , x ∈ X , X n ≥ ( c − m ( λT ) m (cid:18) Sµ (cid:19) m + n ( x ) = X n ≥ ( c − m (cid:18) λµ (cid:19) m (cid:18) Sµ (cid:19) n ( x ) and X c − c m ≤ n ≤ m ( λT ) m (cid:18) Sµ (cid:19) m − n ( x ) = X c − c m ≤ n ≤ m (cid:18) λµ (cid:19) m − n ( λT ) n ( x ) . (cid:3) Remark 2.8. The first two points of Lemma 2.7 tell us that, whenever a T ( X , S )
Let X be a dense subset of X and S : X → X such that T S ( x ) = x forall x ∈ X . Let also ( λ in ) n , i ∈ N , be a countable family of sequences in ( a, b ) . We assumethat(1) There exist c, d ∈ ( a, b ) such that λ in ∈ ( c n , d n ) for any i ∈ N , n ≥ ;(2) There exists C > such that C − λ in + m ≤ λ in λ im ≤ Cλ in + m for any n, m, i ∈ N .Then \ i ∈ N F HC (( λ in T n ) n ) = ∅ . The next proposition tells us that, when X is a Banach space, Theorem 2.6 is not sofar from being optimal. We will see in the next paragraph that it is optimal for a ratherstandard class of weighted shifts. Proposition 2.10. We keep the notations of Theorem 2.6. Let us assume that X is aBanach space. If Λ ⊂ [1 /r ( T ) , + ∞ ) is unbounded or /r ( T ) ∈ Λ , then \ λ ∈ Λ F HC ( λT ) = ∅ . Proof. We only prove the case where /r ( T ) ∈ Λ , the case Λ unbounded being treated verysimilarly. To start with, let us first assume that /r ( T ) is an accumulation point of Λ . Wefix λ ∈ Λ . Upon taking a subsequence, we can assume that Λ = ( λ k ) k ∈ N is decreasing to /r ( T ) . By contradiction, we assume that there exists x ∈ X which is frequently hypercyclicfor all λ k T , k ∈ N . We fix e ∈ X \ { } and denote by N k , k ≥ , the sets respectively givenby N := { n ∈ N : k λ n T n x k < } and N k := { m ∈ N : k λ mk T m x − e k < k e k } , k ≥ . By assumption, d ( N ) ≥ ε > and each N k , k ∈ N , is infinite. So there exists an increasingsequence ( m k ) k ≥ with m k ∈ N k such that m k → + ∞ , and one can define ( n k ) k ≥ by n k := max { n < m k : n ∈ N } . Note that ( m k ) k can be chosen so that ( n k ) k is also increasing and tends to + ∞ . Moreover,from the definition of n k , k ≥ , we get(2.14) ε ≤ d ( N ) ≤ lim sup k card ( N ∩ { , . . . , m k − } ) m k ≤ lim sup k n k m k . Now, by construction, we have for any k ≥ , k T n k x k < λ − n k and λ − m k k k e k < k T m k x k ≤ k T m k − n k kk T n k ( x ) k . It follows, k e k k T m k − n k k > λ n k λ m k k , whence(2.15) (cid:18) λ λ k (cid:19) n k ≤ k e k λ m k − n k k k T m k − n k k ≤ k e k ( λ k T k ) m k − n k . Since ( λ k ) k is decreasing and n k → + ∞ , we first deduce from the last inequality that m k − n k → + ∞ . This gives r ( T ) = lim k k T m k − n k k / ( m k − n k ) . We also derive from (2.15) thefollowing: (cid:18) λ λ k (cid:19) n k /m k ≤ (cid:18) k e k (cid:19) /m k λ − n k /m k k k T m k − n k k /m k , which implies, using that m k → + ∞ and m k − n k → + ∞ , lim sup k n k m k ≤ r ( T ) λ ) lim k (cid:16) ln( λ k k T m k − n k k mk − nk ) (cid:17) = 0 , since by assumption ( λ k ) k is decreasing to /r ( T ) . This contradicts (2.14) and concludesthe proof when /r ( T ) is an accumulation point of Λ .Let us deal with the remaining case, i.e., /r ( T ) ∈ Λ but /r ( T ) is not an accumulationpoint of Λ . We will in fact prove the stronger fact that, if /r ( T ) = λ are both in Λ , then r ( T ) − T and λT share no frequent hypercyclic vectors. The proof goes along the same linesas above. Let us denote µ = 1 /r ( T ) . By assumption λ/µ > . We assume by contradictionthat x is hypercyclic for λT and µT and we set N λ := { n ∈ N : k λ n T n x k < } and N µ := { m ∈ N : k µ m T m x − e k < k e k } , k ≥ . As above, since these sets are infinite, one can define an increasing sequence of integers ( m k ) k ∈ N ⊂ N µ , tending to ∞ , such that the sequence ( n k ) k ∈ N defined by n k := max { n < m k : n ∈ N λ } is increasing. We have d ( N λ ) ≤ lim sup k n k m k and, proceeding exactly as in the first part ofthe proof, m k − n k → ∞ and (cid:18) λµ (cid:19) n k ≤ k e k ( µ k T k ) m k − n k , k ∈ N . Therefore d ( N λ ) ≤ lim sup k n k m k ≤ r ( T ) λ ) lim k (cid:16) ln( µ k T m k − n k k mk − nk ) (cid:17) = 0 , and x is not frequently hypercyclic for λT . (cid:3) Remark 2.11. The proof of the previous proposition tells us a bit more than its statement.More precisely, we have shown that, if Λ is unbounded or if /r ( T ) ∈ Λ , and if x ∈ X isa common hypercyclic vector for all λT , then it can be a frequently hypercyclic vector fornone of the λT . This complements for instance the result saying that the set T λ> HC ( λB ) is different from the set HC ( µB ) for any µ > . Indeed, for µ > , we have F HC ( µB ) ⊂ HC ( µB ) \ \ λ> HC ( λB ) . Another interesting feature of the previous result (more precisely of the one proved inthe second part of the proof) is the idea that it gives to build two frequently hypercyclicoperators (in fact satisfying the Frequent Hypercyclicity Criterion) sharing no frequentlyhypercyclic vectors. This will be detailed in the end of the next Paragraph, see Corollary2.23). OMMON FREQUENT HYPERCYCLICITY 13 In the whole paragraph, we have considered positive real scalar multiples of a single oper-ator T . In virtue of the León-Müller theorem for frequent hypercyclicity ([7, Theorem 6.28]), F HC ( λT ) = F HC ( T ) for any λ ∈ C , | λ | = 1 . It is also known that the Ansari theorem forfrequent hypercyclicity holds true: For any positive integer p , F HC ( T ) = F HC ( T p ) (see[4]). This with Theorem 2.6 thus implies: Corollary 2.12. Let X be a dense set of X and S : X → X such that T S ( x ) = x for all x ∈ X . Let also T be a bounded linear operator on X and Λ ⊂ { z ∈ C : a T ( X , S ) < | z | < b T ( X , S ) } . If {| λ | : λ ∈ Λ } is a countable relatively compact subsetof ( a T ( X , S ) , b T ( X , S )) , then \ λ ∈ Λ F HC ( λT ) = ∅ . If in addition {| λ | /p : p ∈ N ∗ , λ ∈ Λ } is a relatively compact subset of ( a T ( X , S ) , b T ( X , S )) ,then \ λ ∈ Λ , p ∈ N ∗ F HC ( λT p ) = ∅ . Note that the additional assumption above occurs for example if a T ( X , S ) < and b T ( X , S ) = + ∞ (for e.g., for a large class of weighted shifts, see the next paragraph).2.3. Application to weighted shifts.Theorem 2.13. Let X be a Fréchet space with unconditional basis ( e n ) n , and let { w ( λ ) =( w n ( λ )) n , λ ∈ Λ } be a countable family of weights. We assume that there exist a weight ω = ( ω n ) n and constants < η < and M > such that for any λ ∈ Λ and any n ≥ , m ≥ ,(i) The series P n ≥ ( ω . . . ω n ) − e n is convergent in X ;(ii) ω n . . . ω n + m ≤ η m w n . . . w n + m ( λ ) ;(iii) M − m ≤ w n . . . w n + m ( λ ) ≤ M m .Then T λ ∈ Λ F HC ( B w ( λ ) ) is non-empty.Proof. For notational simplicity, let us denote { w ( λ ) = ( w n ( λ )) n , λ ∈ Λ } = ( w ( i )) i ∈ N . Weconsider X = span ( e k : k ≥ 0) = [ n ≥ ker ( T n ) and, for a weight w , the operator F w on X given by F w ( e k ) = 1 w k +1 ( e k +1 ) . By definition of X , we need only check that { B w ( i ) : i ∈ N } satisfies the assumptions (1)–(4)of Corollary 2.4 for any x = e k , k ∈ N . Observe that (2) is trivially satisfied. From now on,for l < , we use the notations e l = 0 and w l ( i ) = 0 , i ∈ N . For any i, j, m, l ∈ N , let us write B mw ( i ) F lw ( j ) ( e k ) = w k + l ( i ) . . . w k + l − m +1 ( i ) w k + l ( j ) . . . w k +1 ( j ) e k + l − m . Note that B nw ( i ) ( e k ) = 0 whenever n is large enough, uniformly for i ∈ N . This gives the firstpart of (1) in Corollary 2.4. Moreover,(2.16) X n ≥ F nw ( i ) ( e k ) = X n ≥ w k + n ( i ) . . . w k +1 ( i ) e k + n By assumption (ii), we have w k + n ( i ) . . . w k +1 ( i ) > ω k + n . . . ω k +1 . So, by assumption (i) andusing that ( e k ) k is an unconditional basis, we get that the left-hand side term in (2.16) is unconditionally convergent in X , uniformly for i , hence the second part of (1) in Corollary2.4.Let us now turn to proving that (3) in Corollary 2.4 holds. By the assumption (i) andunconditionality of ( e k ) k , the sequence ( ω k +1 . . . ω k + n ) − e k + n is bounded uniformly for n ≥ .We denote by k · k the F -norm associated to the Fréchet distance of X . Then, for someconstant K (depending only on k and the constant of unconditionality of ( e k ) k ) and by theassumptions (ii) and (iii), we have k B mw ( i ) F m + nw ( j ) ( e k ) k = k w k + m + n ( i ) . . . w k + n +1 ( i ) w k + m + n ( j ) . . . w k +1 ( j ) e k + n k (2.17) = k w k + m + n ( i ) . . . w k + n +1 ( i ) w k + m + n ( j ) . . . w k + n +1 ( j ) ω k +1 . . . ω k + n w k +1 ( j ) . . . w k + n ( j ) ( ω k +1 . . . ω k + n ) − e k + n k (2.18) ≤ KM m η n . (2.19)So, after writing n = ( c − m + l , l ≥ , we easily check that there exists some c > suchthat M m η ( c − m ≤ for any m ≥ . Since η < , the series P n ≥ ( c − m B mw ( i ) F m + nw ( j ) ( e k ) isabsolutely convergent, uniformly for m ∈ N , which implies (3).The proof of Corollary 2.4, (4) is left to the reader. (cid:3) At this point, we shall make a remark. Remark 2.14. Bayart-Ruzsa [8] proved in 2015 that, when acting on ℓ p spaces, ≤ p < ∞ ,weighted shifts are frequently hypercyclic if and only if they satisfy the Frequent Hyper-cyclicity Criterion (i.e., they are chaotic). This result was extended to more general classesof spaces in [14]. For instance, it is proved there that Bayart-Ruzsa theorem extends to anyBanach space with unconditional basis ( e k ) k whenever ( e k ) k is boundedly complete . We recallthat a basis ( e k ) k in X is called boundedly complete if, for any sequence of scalars ( x k ) k ,whenever the sequence K X k =0 x k e k ! K ≥ is bounded in X , then it is convergent in X . Examples of such Banach spaces are givenamong Köthe sequences spaces (including of course ℓ p spaces). Note that the usual basis of c is not boundedly complete.Anyway, in the situation given by Remark 2.14, Theorem 2.13 can be rephrased as follows. Corollary 2.15. Let X be a Banach space with boundedly complete unconditional basis ( e n ) n , and let { w ( λ ) = ( w n ( λ )) n , λ ∈ Λ } be a countable family of weights. We assume thatthere exist a frequently hypercyclic weighted shift B ω , ω = ( ω n ) n , and constants < η < and M > such that for any λ ∈ Λ and any n ≥ , m ≥ ,(i) ω n . . . ω m + n ≤ η m w n . . . w m + n ( λ ) ;(ii) M − m ≤ w n . . . w m + n ( λ ) ≤ M m .Then T λ ∈ Λ F HC ( B w ( λ ) ) is non-empty. In Fréchet spaces, bounded completeness of the unconditional basis ( e n ) n is not sufficientany more, and some other conditions are given in [14]. As an application, it is shown thaton the space H ( D ) of analytic functions in the unit disc D , endowed with the locally uniformFréchet topology, a weighted shift is frequently hypercyclic if and only if it satisfies theFrequent Hypercyclicity Criterion. Thus the previous corollary holds if the Banach space X is replaced with H ( D ) .Let us give an example. OMMON FREQUENT HYPERCYCLICITY 15 Example 2.16. For λ ∈ (0 , + ∞ ) , let B w ( λ ) be the weighted shift on ℓ ( N ) defined by w n ( λ ) = 1 + λ/n . In [18], it is proven that T λ> HC ( B w ( λ ) ) is residual. We can easilydeduce from Corollary 2.15 that for any countable relatively compact subset Λ of ( , + ∞ ) ,one has \ λ ∈ Λ F HC ( B w ( λ ) ) = ∅ . Furthermore Theorems 2.7 or 2.13 can be applied to a family of multiples of a singleweighted shift. We keep the notations of Paragraph 2.2. We fix a weight w = ( w n ) n andconsider X = span ( e k : k ≥ . First observe that b ( B w , X , F w ) = + ∞ . Let us simplydenote λ w = a ( B w , X , F w ) , that is λ w = inf { λ > the series X λ − n w . . . w n e n is convergent in X } . Note that λ w = lim sup n ( k e n k w ...w n ) /n . We recall that a slight generalization of Abakumov-Gordon theorem states that T λ>λ w HC ( λB w ) is a dense G δ subset of X , see [7, p. 178] or[6]. In this context, Theorem 2.6 reads as follows. Corollary 2.17. Let B w be a weighted shift on a Fréchet space X with unconditional basis ( e n ) n . Then T λ ∈ Λ F HC ( λB w ) is non-empty for any countable relatively compact subset Λ of ( λ w , + ∞ ) . As already said, countability in the previous corollary is necessary. A natural question iswhether it is possible to get common frequent hypercyclicity in the case Λ is bounded but λ w ∈ Λ . We can give partial answers in two different directions. First of all, as mentioned inRemark 2.14, if we additionally assume that X is a Banach space and that the unconditionalbasis ( e n ) n is boundedly complete, then the convergence of the series λ − n w ...w n e n is necessaryfor B w to be frequently hypercyclic [14]. So, in such a case, for any weighted shift B w , \ λ ∈ Λ F HC ( λB w ) = ∅ whenever λ w > inf Λ . Second, for any X with unconditional basis ( e n ) n and any B w suchthat λ w = r − w (where r w denotes the spectral radius r ( B w ) of B w ) \ λ ∈ Λ F HC ( λB w ) = ∅ whenever λ w ∈ Λ (Proposition 2.10).Altogether, these two observations thus give the following: Proposition 2.18. Let X be a separable Banach space with unconditional basis ( e n ) n , w abounded weight and Λ ⊂ [ λ w , + ∞ ) . We assume that ( e n ) n is boundedly complete and that λ w = r − w . Then \ λ ∈ Λ F HC ( λB w ) = ∅ if and only if Λ is countable, bounded and λ w / ∈ Λ .Special case of ℓ p spaces. There of course exist many general situations where the assump-tions of Proposition 2.18 do not hold. However, bounded completeness of course holds when X = ℓ p , ≤ p < ∞ . In this context, it makes sense to examine in which extend the condition λ w = r − w can be relaxed. From now on, p is fixed in [1 , ∞ ) .Observe that for a weighted shift B w acting (boundedly) on ℓ p , one has λ w = 1lim inf( w . . . w n ) /n . It is not difficult to check that λ w = r − p,w with r p,w := r p ( B w ) where r p ( T ) := sup { λ : λ ∈ σ p ( T ) } . Here σ p ( T ) denotes the point spectrum of an operator T (see for e.g., [31, Theorem 8, P.70]). We consider that there cannot be confusion due to the notation p in ℓ p and the other p appearing in r p,w ( T ) (where it is for pointwise ). Thus λ w is in general larger than r − w .Moreover, it is easily seen that, whenever the weight sequence ( w n ) n is bounded, r w = lim n (cid:18) sup k w k . . . w k + n (cid:19) /n = lim sup n (cid:18) sup k w k . . . w k + n (cid:19) /n . Proposition 2.10 then tells us that if Λ ⊂ (1 /r w , ∞ ) is unbounded or admits /r w as an accumulation point, then T λ ∈ Λ F HC ( B w ) = ∅ . Ofcourse, by Corollary 2.17 (or as in Proposition 2.18), this is only interesting when λ w = r − w .This equality happens quite often, but this can also fail to occur (see Example 2.20 below).In fact, for weighted shifts, one can get a little improvement of Proposition 2.10. To stateit, let us introduce the quantity λ w = 1lim sup( w . . . w n ) /n . We observe that r − w ≤ λ w ≤ λ w . Examples of weights w for which r − w = λ w = λ w or r − w = λ w = λ w are easily built (Example 2.20). Corollary 2.19. Let B w be a weighted shift acting on ℓ p and let Λ be a countable subset of [ λ w , + ∞ ) . If Λ is unbounded or λ w ∈ Λ , then \ λ ∈ Λ F HC ( λT ) = ∅ . Proof. It is very similar to that of Proposition 2.10. Let us only give the outline of theproof in the case where Λ is any sequence ( λ k ) k ∈ N decreasing to λ w . By contradiction, let usassume that x = ( x n ) n ∈ N ∈ ℓ p is some frequent hypercyclic vector for each λ k B w , k ∈ N . Asin the proof of Proposition 2.10, we introduce the sets N := { n ∈ N : k λ n B nw x k < } and N k := { m ∈ N : k λ mk B mw x − e k < } , k ≥ . Then, replacing T by B w , we similarly define increasing sequences ( n k ) k ≥ ⊂ N and ( m k ) k ≥ ,tending to + ∞ , with m k ∈ N k and such that d ( N ) ≤ lim sup k n k /m k and, for any k ≥ , λ n k w m k − n k +1 . . . w m k x m k < λ m k k w . . . w m k x m k > . It follows λ n k λ m k k < w . . . w m k − n k ; In particular m k − n k → + ∞ and ( λ /λ k ) n k /m k < /m k λ − n k /m k k ( w . . . w m k − n k ) /m k hence lim sup k n k m k ≤ C (lim sup k ln( λ k ) − ln( λ w )) = 0 , for some constant C ≥ . This contradicts d ( N ) > . (cid:3) Let us provide with some examples. OMMON FREQUENT HYPERCYCLICITY 17 Example 2.20. 1) For the class W of bounded weights w satisfying λ w = λ w , then Corol-laries 2.17 and 2.19 give the following necessary and sufficient condition: Proposition 2.21. Let w ∈ W and Λ ⊂ [ λ w , + ∞ ) . The set T λ ∈ Λ F HC ( λB w ) is non-emptyif and only if Λ is countable, bounded and λ w / ∈ Λ . The class W contains, for e.g., all weights w = ( w n ) n such that ( w n ) n is convergent. Yet,for such weights, we even have λ w = r − w and Proposition 2.18 applies.It is in fact not difficult to provide with a set of examples of weights w which allows todistinguish all the quantities k B w k − , r − w , λ w and λ w . Indeed, let a ≤ b ≤ c ≤ d be fourpositive real numbers, and let us define w n := a if n ∈ { , . . . , } ∪ { k ( k − , . . . , k − } d if n = 2 k c if n ∈ { k + 1 , k + k + 1 } b if n ∈ { k + k + 2 , ( k + 1)2 k } , k ≥ . Then one easily checks that k B w k − = 1 /d ≤ r − w = 1 /c ≤ λ w = 1 /b ≤ λ w = 1 /a. Therefore, if we choose a = b = c , then Proposition 2.21 tells us that T λ ∈ Λ F HC ( λB w ) isnon-empty if and only if Λ is countable, bounded and λ w / ∈ Λ . Yet Proposition 2.18 cannotbe used here.In view of the previous discussion, it seems to be reasonable to wonder whether /r ( T ) could be replaced by /r p ( T ) in Proposition 2.10. More precisely, for the weighted shifts on ℓ p , one can pose the following: Question 2.22. Do there exist weighted shifts B w on ℓ p , ≤ p < ∞ , with λ w < λ w andsome countable Λ ⊂ ( λ w , + ∞ ) such that λ w ∈ Λ and T λ ∈ Λ F HC ( λB w ) is non-empty? We conclude the paragraph by showing how the results of this paragraph permit to easilyexhibit two (or more) explicit frequently hypercyclic weighted shifts which share no frequentlyhypercyclic vector. We can then state the following. Corollary 2.23. There exist two frequently hypercyclic weighted shifts on ℓ p , ≤ p < + ∞ (hence satisyfing the Frequent Hypercyclicity Criterion), with no common frequent hypercyclicvector.Proof. Let w n = ( n +1 n ) . Since ( w n ) n is decreasing to , one has r − w = λ w = λ w = 1 .Moreover, B w is frequently hypercyclic, since P n ≥ ( w . . . w n ) − < ∞ . Thus, applyingProposition 2.21 with Λ = { , λ } , λ > , we get F HC ( B w ) ∩ F HC ( λB w ) = ∅ . (cid:3) Other examples. In this paragraph, we apply our general common frequent hy-percyclicity criteria to classical frequently universal sequences of operators which are notweighted shifts.Since almost all the classical examples of frequent hypercyclic operators satisfy the Fre-quent Hypercyclicity Criterion, the range of applications of Theorem 2.6 is almost the largest.Let us give one example. Example 2.24 (Differential operators on H ( C ) ) . Let D be the differentiation operator on H ( C ) , Df ( z ) = f ′ ( z ) . Costakis and Mavroudis showed [17] that for any non-constant poly-nomial P , P ( D ) satisfies the Bayart-Matheron criterion (Theorem 2.5) with a P ( D ) ( X , S ) = 0 and b P ( D ) ( X , S ) = + ∞ for some dense subset X of X and some right inverse S of P ( D ) on X . Thus, with the frequent hypercyclicity version of the León-Müller theorem and Theorem2.6, we can deduce that \ λ ∈ Λ F HC ( λP ( D )) = ∅ , for any countable relatively compact subset Λ of C ∗ .We shall now focus on applications of Theorem 2.3 to families of operators which are notmultiples of a single one. Example 2.25 (Adjoint of multipliers on the Hardy space) . We denote by D := { z ∈ C : | z | < } the unit disc, by H ∞ the space of bounded analytic functions in D , and by H theclassical Hardy space, H := ( f = X k ≥ a k z k ∈ H ( D ) : k f k := ( X k ≥ | a k | ) / < ∞ ) . We recall that H ∞ and H are Banach spaces, endowed respectively by the sup -norm k · k ∞ and the k · k . Let Φ ∈ H ∞ and Φ ∗ ∈ L ∞ ( T ) its boundary value. Let us assume that Φ isnot outer and that / Φ ∈ H ∞ . We denote by M Φ : H → H the multiplication operatorwith symbol Φ , M Φ ( f ) = Φ f , and by M ∗ Φ its adjoint. It is known [7] that λM ∗ Φ is frequentlyhypercyclic on H for any λ > k / Φ k ∞ and that \ λ> k / Φ k ∞ HC ( λM ∗ Φ ) is a G δ -subset of H [24].Now, let us write the inner-outer decomposition Φ = uθ , with u outer and θ the non-constant inner part of Φ . Let us define X := ∪ n ≥ K n with K n := H ⊖ θ n H . Then X isthe generalized kernel of M ∗ Φ and is dense in X . Moreover, if we define S := M ∗ /u M θ , then M ∗ Φ S = Id and k S k = k / Φ k ∞ . We refer, for e.g., to the proof of [24, Theorem 3.1] for thedetails concerning the previous claims. So, with the notations introduced before Theorem2.6, we have a ( M ∗ Φ , X , S ) ≤ k S k = k / Φ k ∞ and b ( T, X , S ) = + ∞ . Therefore, Theorem2.6 directly implies that \ λ ∈ Λ F HC ( λM ∗ Φ ) = ∅ , whenever Λ is a countable relatively compact subset of ( k / Φ k ∞ , + ∞ ) .In fact, we can deduce from Corollary 2.4 the following more general result. Proposition 2.26. Let { Φ λ : λ ∈ Λ } be a countable family of bounded analytic functionson D with the same non-constant inner factor θ . We assume that a := sup {k Φ − λ k ∞ : λ ∈ Λ } < M := sup {k Φ λ / Φ µ k ∞ : λ ∈ Λ } < ∞ . Then \ λ ∈ Λ F HC ( M ∗ φ λ ) = ∅ . Proof. We aim to apply Corollary 2.4. By the comment after its statement, we need onlycheck items (2)–(4). Since the functions Φ λ share the same non-constant inner factor, the set X := ∪ n ≥ K n with K n := H ⊖ θ n H is the generalized kernel of each M ∗ φ λ . Let u λ denotethe outer factor of Φ λ . As recalled above, setting S λ := M ∗ /u λ M θ , we have T nλ S nλ = Id for OMMON FREQUENT HYPERCYCLICITY 19 any n ∈ N . So (2) and (4) of Corollary 2.4 are satisfied. Let λ = µ ∈ Λ and f ∈ X . Byassumption, there exists b ∈ ( a, such that for any m ∈ N , writing n = ( c − m + k , k ∈ N , k T mλ S m + nµ ( f ) k = sup k g k =1 (cid:10) T mλ S m + nµ ( f ) , g (cid:11) = sup k g k =1 (cid:28) f, (cid:18) u λ u µ (cid:19) m (cid:18) ¯ θu µ (cid:19) n g (cid:29) ≤ k f k (cid:13)(cid:13)(cid:13)(cid:13) u λ u µ (cid:13)(cid:13)(cid:13)(cid:13) m ∞ (cid:13)(cid:13)(cid:13)(cid:13) u µ (cid:13)(cid:13)(cid:13)(cid:13) n ∞ ≤ k f k (cid:0) M ( a/b ) ( c − (cid:1) m (cid:16) ab (cid:17) k . Since a/b < , (3) of Corollary 2.4 then follows by taking c > such that M ≤ ( b/a ) ( c − . (cid:3) Periodic points at the service of common frequent hypercyclicity Despite its apparent unpleasant formulation, the classical Frequent Hypercyclicity Crite-rion turns out to be very useful for checking that natural operators are frequently hypercyclic(and chaotic). We saw in the previous section that it fits well to formulating easy-to-use suf-ficient conditions for common frequent hypercyclicity. In [25], the authors provided a quiteappealing new criterion for frequent hypercyclicity and chaos involving the periodic pointsof the operator [25, Theorem 5.31]. It is shown there that all the operators which satisfythe Frequent Hypercyclicity Criterion satisfy the assumptions of this new one. However, itquickly appears from its statement that it is not so simple to use when dealing with naturaloperators (for e.g., weighted shifts). Yet it is very well adapted to certain type of operatorswhich were introduced by Menet in [27] to build chaotic operators on ℓ p which are not fre-quently hyperyclic. These operators have been extensively developed - and called operatorsof C-type - in [25, Section 6] to build several counter-examples.In this section, we provide with a sufficient condition for common frequent hypercyclicityderived from [25, Theorem 5.31]. We recall that a vector x ∈ X is a periodic point for T ∈ L ( X ) if there exists p ∈ N such T p x = x . Let us denote by Per ( T ) the set of all periodicpoints for T . For x ∈ Per ( T ) we denote by p T ( x ) the period of x for T (i.e., the smallestpositive integer p such that T p x = x ). Theorem 3.1. Let { T s : s = 1 , , , . . . } be a countable family in ∈ L ( X ) . We assume thatthere exists a dense linear subspace X of X with T s ( X ) ⊂ X and X ⊂ Per ( T s ) for any s ≥ , and a constant α ∈ (0 , such that the following property holds true: For every s ≥ ,every ε > , every x, y ∈ X , every q ≥ and every t , . . . , t q ≥ , there exist z ∈ X andintegers n, d ≥ such that(1) d is a multiple of p T ti ( y ) and of p T ti ( z ) for each i = 1 , . . . , q ;(2) k T kt z k < ε for every ≤ k ≤ αd and every t ≥ ;(3) k T n + ks z − T ks x k < ε for every ≤ k ≤ αd .Then there exists a vector in X which is frequently hypercyclic for each T s , s ≥ . If the family { T s : s ∈ N \ } is reduced to a single operator, Theorem 3.1 is exactly [25,Theorem 5.31]. Yet one should mention that the previous statement does not only mean"each T s satisfies the assumptions of [25, Theorem 5.31]". It would be interesting to knowwhether two operators satisfying the assumptions of [25, Theorem 5.31] automatically havea frequently hypercyclic vector in common. Note that we already saw that two operators T and T may have no common frequently hypercyclic vector, even if they both satisfy theclassical Frequent Hypercyclicity Criterion (see Corollary 2.23). Finally note that Theorem ( T ) ∩ Per ( λT ) = ∅ in general. Proof of Theorem 3.1. As one could expect, the proof is greatly inspired by that of [25,Theorem 5.31]. Let ( x l ) l ≥ be a sequence of vectors in X , dense in X , and let ( I p ( s )) p,s ≥ be a partition of N such that each set I p ( s ) is infinite and has bounded gaps. Let us denoteby r p ( s ) the maximal size of a gap for I p ( s ) . We also let ( y j ) j ∈ N be given by y j = x p if j ∈ I p ( s ) . Now we use the assumptions of the theorem to build, by induction on j ∈ N asequence ( z j ) j ∈ N of vectors in X and increasing sequences of positive integers ( d j ) j ∈ N and ( n j ) j ∈ N such that the following properties hold, if j ∈ I p ( s ) :(i) d j is a multiple of p T t ( P j − k =1 z k ) and p T t ( z j ) for every t so that there exist q ≥ and ≤ i ≤ j with i ∈ I q ( t ) ;(ii) k T kt ( z j ) k < − j for every ≤ k ≤ αd j and every t ≥ ;(iii) k T n j + ks z j − T ks ( y j − P j − i =1 z i ) k < − j for every ≤ k ≤ αd j ;(iv) n j is a multiple of p T s ( P j − i =1 z i ) and αd j < n j ≤ d j ;(v) αd j > d j − .By (ii), the sum z := P i ≥ z i defines a vector in X . Let us check that z is frequentlyhypercyclic for every T s , s ≥ .Let p, s ≥ be fixed. We set I p ( s ) := { j m : m ≥ } , where ( j m ) m ≥ is increasing andsatisfies j m +1 − j m ≤ r p ( s ) for every m ≥ . Then, for every m ≥ we define by inductionon j ∈ N a family of sets ( A m,j,s ) ≤ j To conclude we now turn to proving that for every n ∈ A m,j,s (3.1) k T ns j m + j X i =1 z i ! − x p k ≤ j X i =0 − ( j m + i ) . To do so, we proceed by induction on ≤ j < j m +1 − j m . If n ∈ A m, ,s , then j = 0 and n = n j m + kd j m + k ′ p T s ( x p ) with ≤ k ≤ αd jm +1 d jm − and ≤ k ′ ≤ αd jm p Ts ( x p ) and by (i) and (iv) T ns j m X i =1 z i ! − x p = T n jm + kd jm + k ′ p Ts ( x p ) s j m X i =1 z i ! − x p = T n jm + k ′ p Ts ( x p ) s ( z j m ) − T k ′ p Ts ( x p ) s x p − j m − X i =1 z i ! . By (iii) we get k T ns j m X i =1 z i ! − x p k ≤ − j m . Assume now that (3.1) has been proven up to j − for some ≤ j < j m +1 − j m . For n ∈ A m,j,s , we write n = kd j m + j + l with l ∈ A m,j − ,s and ≤ k ≤ αd j m + j +1 d j m + j − . Then, by (i) we have T ns j m + j X i =1 z i ! − x p = T kd jm + j + ls j m + j X i =1 z i ! − x p = T ls j m + j − X i =1 z i ! − x p + T ls ( z j m + j ) Since l ∈ A m,j − ,s , we deduce from the induction hypothesis and (ii) that k T ns j m + j X i =1 z i ! − x p k ≤ j − X i =0 − ( j m + i ) + 2 − ( j m + j ) , and (3.1) as desired. (cid:3) Application to operators of C -type. We will apply Theorem 3.1 to operators of C -type on ℓ p ( N ) . First we shall recall their definition, following the formalism of [25, Section 6]. Asusual, we denote by ( e k ) k ∈ N the canonical basis of ℓ p ( N ) . An operator of C -type is associateda data of four parameters v, w, ϕ and b : • v = ( v n ) n ≥ is a sequence of non-zero complex numbers with P n ≥ | v n | < ∞ ; • w = ( w n ) n ≥ is a sequence of complex numbers such that < inf n ≥ | w n | ≤ sup n ≥ | w n | < ∞ ; • ϕ : N → N is such that ϕ (0) = 0 , ϕ ( n ) < n for every n ≥ , and the set { n ∈ N : ϕ ( n ) = l } is infinite for every l ≥ ; • b = ( b n ) n ≥ is a strictly increasing sequence of positive integers with b = 0 and b n +1 − b n is a multiple of b ϕ ( n )+1 − b ϕ ( n ) ) for every n ≥ . Now, for a data v, w, ϕ and b as above, the operator of C -type T v,w,ϕ,b is defined by T v,w,ϕ,b e k = w k +1 e k +1 if k ∈ [ b n , b n +1 − , n ≥ v n e b ϕ ( n ) − (cid:16)Q b n +1 − j = b n +1 w j (cid:17) − e b n if k = b n +1 − , n ≥ − (cid:16)Q b − j = b +1 w j (cid:17) − e if k = b − . Here, by convention, en empty product is equal to . From now on, we assume that thecondition inf n ≥ Y b n Let { T v ( s ) ,w ( s ) ,ϕ,b : s ∈ N } be a countable family of operators of C + -type on ℓ p ( N ) where b does not depend on s . We assume that there exists a constant α > such thatfor every s ≥ , every C ≥ and every k ≥ , there exists an integer k ≥ k such that, forevery ≤ n ≤ α ∆ ( k ) , (3.2) | v ( k ) ( s ) | ∆ ( k ) − Y i = n +1 | w ( k ) i ( s ) | > C. If there exists a constant K > such that for any s, t ≥ and any r ≥ p ≥ , (3.3) (cid:12)(cid:12)(cid:12)(cid:12) w p ( s ) w p +1 ( s ) . . . w r ( s ) w p ( t ) w p +1 ( t ) . . . w r ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K, then T s ≥ F HC ( T v ( s ) ,w ( s ) ,ϕ,b ) is non-empty. Note that since b does not depend on s , by definition the ∆ ( k ) , k ≥ , do not depend on s either. It is plainly checked that condition (3.2) is equivalent to saying that each T s satisfiesthe assumption of [25, Theorem 6.9]. In particular, if { T v ( s ) ,w ( s ) ,ϕ,b : s ∈ N } is reduced toa single operator (i.e., v ( s ) and w ( s ) do not depend on s ), then the previous criterion isexactly [25, Theorem 6.9].For the proof of Theorem 3.2, we recall [25, Fact 6.8] below. OMMON FREQUENT HYPERCYCLICITY 23 Fact 1. Let T be an operator of C + -type on ℓ p ( N ) and k ≥ . For any l < k − and ≤ m ≤ ∆ ( k ) , we have T m e b k − l +1 − m = v ( k ) ∆ ( k ) − Y i =∆ ( k ) − m +1 w ( k ) i e b l − ∆ ( k ) − m Y i =1 w ( k ) i − e b k − l . Proof of Theorem 3.2. Without loss of generality, we can assume that < α < . It sufficesto check that the assumptions of Theorem 3.1 are satisfied. Let us define X := span ( e k ; k ∈ N ) and fix x, y ∈ X , ε > and s ≥ . There exists k ≥ such that x = X l< k b l +1 − X j = b l x j e j . By (3.2), for any C > , there exists k ≥ k such that | v ( k ) ( s ) | ∆ ( k ) − Y i = n +1 | w ( k ) i ( s ) | > C, ≤ n ≤ α ∆ ( k ) . Since v ( s ) and w ( s ) are bounded, upon choosing C large enough, we may assume that k isso large that the following holds true:(a) ∆ ( k ) is a multiple of p T t ( y ) for any t ≥ ;(b) ∆ ( k ) ≤ min((1 − α )∆ ( k ) , α ∆ ( k ) − .Note that, by the definition of b and ϕ for operators of C + -type, and since the period ofany vector in X depends only on the sequence b , (a) is satisfied whenever y is supported in [0 , b k − [ . Let us now set n := ∆ ( k ) − , d := 2∆ ( k ) and z := X l< k b l +1 − X j = b l x j v ( k ) ( s ) ∆ ( k ) − Y i = j − b l +2 w ( k ) i ( s ) − j − b l Y i =1 w b l + i ( s ) ! − e b k − l +1 − n + j − b l . Like for (a) above, d is a multiple of p T s ( z ) for any s ≥ . Thus condition (1) of Theorem3.1 is satisfied.Let us now fix ≤ m ≤ αd and t ≥ . We observe that for every l < k and b l ≤ j ≤ b l +1 − , we have b k − + l +1 − n + j − b l + m ∈ [ b k − + l , b k − + l +1 ) . Indeed, by definition b k − + l +1 − b k − + l = ∆ ( k ) and by (b), − ∆ ( k ) ≤ − n + j − b l + m ≤ .So for every t ≥ , we have T mt e b k − l +1 − n + j − b l = ∆ ( k ) − n + j − b l + m Y i =∆ ( k ) − n + j − b l +1 w ( k ) i ( t ) e b k − l +1 − n + j − b l + m , hence the expression(3.4) T mt ( z ) = X l< k b l +1 − X j = b l x j v ( k ) ( s ) ∆ ( k ) − Y i = j − b l + m +2 w ( k ) i ( s ) − j − b l Y i =1 w b l + i ( s ) ! − j − b l + m +1 Y i = j − b l +2 w ( k ) i ( t ) w ( k ) i ( s ) ! e b k − l +1 − n + j − b l + m . Using (b), we know that ≤ j − b l + m + 1 ≤ α ∆ ( k ) which, by (3.2), (3.3) and the definitionof C + -type operators, implies that for some constant A > (independent of k ), k T mt ( z ) k ≤ k x k C − KA ∆ ( k . Up to choose C large enough, we get (2) in Theorem 3.1.Let us now estimate the norm of T n + ms z − T ms ( x ) for ≤ m ≤ αd . By Fact 1, we obtain T n − ( j − b l ) s e b k − l +1 − n + j − b l = v ( k ) ( s ) ∆ ( k ) − Y i =∆ ( k ) − n + j − b l +1 w ( k ) i ( s ) e b l − ∆ ( k ) − n + j − b l Y i =1 w ( k ) i ( s ) − e b k − l . Applying T j − b l s yields T ns e b k − l +1 − n + j − b l = v ( k ) ( s ) ∆ ( k ) − Y i = j − b l +2 w ( k ) i ( s ) j − b l Y i =1 w b l + i ( s ) ! e j − (cid:16) w ( k ) j − b l +1 ( s ) (cid:17) − e b k − l + j − b l . Moreover, since m + j − b l < ∆ ( k ) , we have T ms e b k − l + j − b l = j − b l + m Y i = j − b l +1 w ( k ) i ( s ) ! e b k − l + j − b l + m , hence T n + ms e b k − l +1 − n + j − b l = v ( k ) ( s ) ∆ ( k ) − Y i = j − b l +2 w ( k ) i ( s ) j − b l Y i =1 w b l + i ( s ) ! T ms e j − (cid:16) w ( k ) j − b l +1 ( s ) (cid:17) − j − b l + m Y i = j − b l +1 w ( k ) i ( s ) ! e b k − l + j − b l + m . By definition of z , it follows that T n + ms ( z )= T ms ( x ) − X l< k b l +1 − X j = b l x j v ( k ) ( s ) ∆ ( k ) − Y i = j − b l + m +1 w ( k ) i ( s ) − j − b l Y i =1 w b l + i ( s ) ! − e b k − l + j − b l + m By assumption, we thus get k T n + ms ( z ) − T ms ( x ) ≤ k x k C − A ∆ ( k and condition (3) of Theorem 3.1 with α ′ = α , as desired. (cid:3) Remark 3.3. 1) It is clear from the proof that the conclusion of Theorem 3.2 remains trueif condition (3.3) is replaced by the following weaker (but less nice) one:(3.5) sup t ≥ ≤ j< ∆ ( k ≤ m ≤ α ∆ ( k ) | v ( k ) ( s ) | ∆ ( k ) − Y i = j +2 | w ( k ) i ( s ) | − j + m +1 Y i = j +2 | w ( k ) i ( t ) | ! < C OMMON FREQUENT HYPERCYCLICITY 25 2) Moreover, if (3.3) is replaced by the following two conditions:(a) For some A > , sup i,t ≥ max (cid:16) | w i ( t ) | ; | w i ( t ) | (cid:17) ≤ A ,(b)(3.6) sup t ≥ ≤ m ≤ α ∆ ( k ) | v ( k ) ( s ) | ∆ ( k ) − Y i =1 | w ( k ) i ( s ) | − m +1 Y i =1 | w ( k ) i ( t ) | ! < C then it is also clear that (3.5) holds, and the conclusion of Theorem 3.2 is still true.It turns out that for a certain subclass of operators of C + -type, for which (2a) holds true,some rather simple condition for frequent hypercyclicity is given in [25]. We shall now seethat a similar condition for a family of operators in this subclass implies (3.6) and thuscommon frequent hypercyclicity. Application to operators of C + , -type. Operators of C + , -type are introduced in [25, Section6.5] as those for which the parameters v and w satisfy the following extra condition: Forevery k ≥ , v ( k ) = 2 − τ ( k ) and w ( k ) i = (cid:26) if ≤ i ≤ δ ( k ) if δ ( k ) < i < ∆ ( k ) , where τ := ( τ ( k ) ) k ≥ and δ := ( δ ( k ) ) k ≥ are two strictly increasing sequences of integers suchthat δ ( k ) < ∆ ( k ) , k ≥ . Within this class of operators of C + , -type, that we simply denoteby T τ,δ,ϕ,b , examples of frequently hypercyclic operators which are not ergodic were providedin [25]. Theorem 3.4. Let { T τ ( s ) ,δ ( s ) ,ϕ,b : s ≥ } be a countable family of operators of C + , -type on ℓ p ( N ) where b does not depend on s . If inf t ≥ lim sup k →∞ δ ( k ) ( t ) − τ ( k ) ( t )∆ ( k ) > , then T s ≥ F HC ( T τ ( s ) ,δ ( s ) ,ϕ,b ) is non-empty.Proof. Remark that (2a) in Remark 3.3 trivially holds, thus it is enough to check (2b). Todo so, we define α < min (cid:18) , 12 inf t ≥ lim sup k →∞ δ ( k ) ( t ) − τ ( k ) ( t )∆ ( k ) (cid:19) . Let s, k ≥ and C ≥ , and let us set n = ∆ ( k ) − . Since ∆ ( k ) → ∞ as k → ∞ , thereexists k ≥ k such that: δ ( k ) ( s ) − τ ( k ) ( s )∆ ( k ) > α and α ∆ ( k ) > ln ( C ) . Then it follows from the definition of operators of C + , -type that (2b) in Remark 3.3 isequivalent to τ ( k ) ( s ) − δ ( k ) ( s ) sup t ≥ ≤ m ≤ α ∆ ( k ) m +1 Y i =1 | w ( k ) i ( t ) | < C . Now, we have sup t ≥ ≤ m ≤ α ∆ ( k ) m +1 Y i =1 | w ( k ) i ( t ) | ≤ α ∆ ( k ) ≤ ( δ ( k ) ( s ) − τ ( k ) ( s )) . Hence, τ ( k ) ( s ) − δ ( k ) ( s ) sup t ≥ ≤ m ≤ α ∆ ( k ) m +1 Y i =1 | w ( k ) i ( t ) | ≤ ( τ ( k ) ( s ) − δ ( k ) ( s )) < − α ∆ ( k ) < C It remains to check that for every ≤ n ≤ α ∆ ( k ) , | v ( k ) ( s ) | ∆ ( k ) − Y i = n +1 | w ( k ) i ( s ) | > C which works the same as in the proof of [25, Theorem 6.17]. (cid:3) Remark 3.5. When one considers only a single operator, Theorem 3.4 is exactly [25, The-orem 6.17].4. Common frequent hypercyclicity with respect to densities We refer to [23] for the abstract definitions and the study of generalized lower/upperdensities . In particular it is proven there that to any sequence of non-negative real numbers α such that P k ≥ α k = + ∞ , one can associate generalized lower and upper densities d α and d α by the formulae d α ( E ) = lim inf n X k ≥ α n,k F ( k ) and d α ( E ) = 1 − d α ( N \ E ) , E ⊂ N , where ( α n,k ) n,k ≥ is the matrix given by α n,k = (cid:26) α k / ( P nj =1 α j ) for ≤ k ≤ n, otherwise.Then we also have d α ( E ) = lim sup n P + ∞ k =1 α n,k F ( k ) . Let us also introduce the notation ϕ α for the function defined by ϕ α ( x ) = P k ≤ x α k , x ∈ [0 , + ∞ ) .For α and β two sequences as above, let us write α . β if there exists k ∈ N such that ( α k /β k ) k ≥ k is decreasing to . As recalled in the introduction, we have d β ( E ) ≤ d α ( E ) ≤ d α ( E ) ≤ d β ( E ) , E ⊂ N , whenever α . β (see [20, Lemma 2.8]). Thus one can define scales of well-ordered densitieswith respect to the type of growth of the defining sequences. In this section, two types ofsequences will play an important role.(1) For ≤ ε ≤ , E ε := (exp( k ε )) k ≥ . By a summation by parts, one can see that for < ε < , ϕ E ε ( n ) ∼ n − ε ε exp( n ε ) (where u k ∼ v k means u k /v k → );(2) For s ∈ N ∪ {∞} , D s := (exp( k/ log ( s ) ( k ))) k ≥ where log ( s ) = log ◦ · · · ◦ log , log appearing s times, with the conventions log (0) ( x ) = x and log ( ∞ ) ( x ) = 1 for any x > . One can check that ϕ D s ( n ) ∼ log ( s ) ( n ) exp( x/ log ( s ) ( n )) for s ∈ N (see [21,Remark 3.10]) and ϕ D ∞ ( n ) ∼ ee − exp( n ) .For r ≥ we shall also write P r := ( k r ) k ≥ . More examples of generalized densities canbe found in [20, 21]. Observe that the usual lower density d (associated to the constantsequence (1 , , , . . . ) ) corresponds to d E , d D and d P . Note that E shall simply be denotedby E and d E = d D ∞ by d E . For any < δ ≤ ε ≤ , any s ≤ t ∈ N and any r ≥ , we thushave d E ≤ d D t ≤ d D s ≤ d E ε ≤ d E δ ≤ d P r ≤ d ≤ d ≤ d P r ≤ d E δ ≤ d E ε ≤ d D s ≤ d D t ≤ d E . OMMON FREQUENT HYPERCYCLICITY 27 As for ( U -)frequently hypercyclic operators, we now say that T ∈ L ( X ) is α -frequentlyhypercyclic (resp. U α -frequently hypercyclic) if there exists x ∈ X such that for any non-empty open set U in X , d α ( N ( x, U, T )) (resp. d α ( N ( x, U, T )) ) is positive. We denote by F HC α ( T ) (resp. U F HC α ( T ) ) the set of all α -frequently (resp. U α -frequently) hypercyclicvectors for T . As proven in [20], no operator can be E -frequently hypercyclic (and hence α -frequently hypercyclic whenever E . α ).A first natural question arises: Question 4.1. Does the work done in Section 2 extend to α -frequent hypercyclicity for some α ? Let us recall that any operator satisfying the Frequent Universality Criterion is automat-ically α -universal whenever α . D s for some s ≥ [21]. Since each of the criteria givenin Section 2 are natural strengthenings of the Frequent Hypercyclicity Criterion, we couldexpect a positive answer to this question for any such α . Moreover, it is easily seen that F HC P r ( T ) = F HC ( T ) for any r ≥ (see [20, Lemma 2.10]), so Question 4.1 has at leastan obvious positive answer for sequences with polynomial growth.Yet the next proposition shows that Theorem 2.6 for the multiples of a single operatorcompletely fails to extend to α -frequent hypercyclicity as soon as E ε . α for some ε > .We will denote by B ( a, r ) the open ball centered at a with radius r . Proposition 4.2. Let < ε ≤ , T ∈ L ( X ) and < λ = µ < + ∞ . Then for any x ∈ HC ( µT ) ∩ HC ( λT ) and any r > ,(1) If µ > λ , then d E ε ( N ( x, B (0 , r ) , λT )) = 1 ;(2) If λ > µ , then d E ε ( N ( x, B (0 , r ) , λT )) = 0 .In particular, HC ( µT ) ∩ F HC E ε ( λT ) = ∅ .Proof. Of course, we shall assume that HC ( λT ) ∩ HC ( µT ) = ∅ . Throughout the proof, r > is fixed. We first prove (1). By assumption, there exists an increasing sequence ( p k ) k ∈ N ⊂ N such that k µ p k T p k x k < r for any k ∈ N . Writing λ p k + i T p k + i x = λ i T i (cid:18) λµ (cid:19) p k µ p k T p k x, i ∈ N , we easily check that k λ p k + i T p k + i x k < r whenever ( λ k T k ) i < ( µ/λ ) p k . Since by assumption λT is hypercyclic, we have λ k T k > . Thus there exists a constant C > (depending on λ, µ and T , but not on k ) such that for any i < Cp k , k λ p k + i T p k + i x k < r . Therefore, [ k ∈ N { p k , . . . , ⌊ (1 + C ) p k ⌋} ⊂ N ( x, B (0 , r ) , λT ) . It follows that d E ε ( N ( x, B (0 , r ) , λT )) ≥ − lim k (cid:18) ϕ E ε ( p k ) ϕ E ε ((1 + C ) p k ) (cid:19) = 1 . (2) is proved similarly. Since x ∈ HC ( µT ) , there exists an increasing sequence ( p k ) k ∈ N ⊂ N such that k µ p k T p k x k > r . Writing T i λ p k − i T p k − i = λ − i ( λ/µ ) p k µ p k T p k , ≤ i ≤ p k , one caneasily check that k λ p k − i T p k − i x k ≥ r ( λ k T k ) − i ( λ/µ ) p k , ≤ i ≤ p k . Thus k λ p k − i T p k − i x k > r whenever ( λ k T k ) i > ( λ/µ ) p k . Since µ k T k > , the last inequality is equivalent to i ∈{⌊ Cp k ⌋ + 1 , . . . , p k } for some constant < C < not depending on k . Therefore, d E ε ( N ( x, X \ B (0 , r ) , λT )) ≥ − lim k (cid:18) ϕ E ε ( ⌊ Cp k ⌋ + 1) ϕ E ε ( p k ) (cid:19) = 1 . (cid:3) Remark 4.3. 1) A trivial argument shows that any hypercyclic operator is automatically U E -frequently hypercyclic. This comes from the fact that any infinite subset of N has positiveupper E -density. Indeed, if E = ( n k ) k ∈ N is an increasing sequence, then d E ( E ) ≥ lim k (cid:18) e n k ϕ E ( n k ) (cid:19) = 1 − e . At the opposite, it turns out that for any sequence of non-negative real numbers α = ( α k ) with P α k = + ∞ such that α n / ( P nk =1 α k ) tends to as n tends to infinity, there exists ahypercyclic operator T ∈ L ( X ) which is not U α -frequently hypercyclic. This can be deducedfrom the fact that for such α , U α -frequent hypercyclicity implies reiterative hypercyclicity [22] and that reiteratively hypercyclic weighted shifts on ℓ p , ≤ p < + ∞ are automaticallyfrequently hypercyclic [10] (and, of course, some hypercyclic weighted shifts are not fre-quently hypercyclic). For the definition of reiterative hypercyclicity, we refer to [10]. Thus,in particular, this observation applies to the weights E ε for all ≤ ε < and D s for all s ∈ N . 2) The algebraic approach to common (frequent) hypercyclicity mentioned in the intro-duction is still efficient when dealing with α -frequent hypercyclicity. In particular, the sameproof as that of [7, Theorem 6.28] shows that for any weight sequence α of non-negative realnumbers satisfying P k ∈ N α k = + ∞ and such that α n / ( P nk =1 α k ) decreases to 0 as n goes toinfinity and any λ with modulus , F HC α ( λT ) = F HC α ( T ) .Indeed, it suffices to follow the same lines as in the proof for frequent hypercyclic operatorsthat one may find in [7] replacing Lemma 6.29 by the following. Lemma 4.4. Let A ⊂ N have positive lower α -density with a non-decreasing weight sequence ( α k ) k ∈ N so that P k ∈ N α k = + ∞ and α n / ( P nk =1 α k ) decreases to zero as n tends to infinity.Let also I , . . . , I q ⊂ N be such that ∪ qj =1 I j = N and n , . . . , n q ∈ N . Then, B := ∪ qj =1 ( n j + A ∩ I j ) has positive lower α -density.Proof. Let N := max ≤ i ≤ q ( n i ) . Then, for any M ≥ N , P M + Nk =1 α k B ( k ) P M + Nk =1 α k ≥ q P qj =1 P M + Nk =1 α k n j + A ∩ I j ( k ) P M + Nk =1 α k ≥ q P qj =1 P Mk =1 α k A ∩ I j ( k ) P M + Nk =1 α k since α k + n j ≥ α k ≥ q P Mk =1 α k A ( k ) P M + Nk =1 α k = 1 q P Mk =1 α k A ( k ) P Mk =1 α k P Mk =1 α k P M + Nk =1 α k On the other hand, P Mk =1 α k P M + Nk =1 α k = 1 − P M + Nk = M +1 α k P M + Nk =1 α k ≥ − M + N X j = M +1 α j P jk =1 α k ! −→ M →∞ . Hence, d α ( B ) = lim inf M →∞ P M + Nk =1 α k B ( k ) P M + Nk =1 α k ≥ lim inf M →∞ q P Mk =1 α k A ( k ) P Mk =1 α k = 1 q d α ( A ) > . (cid:3) We conclude by an answer to the following natural question: OMMON FREQUENT HYPERCYCLICITY 29 Question 4.5. Does there exist an operator which is α -frequently hypercyclic for any α . E ? This question is clearly a question of common frequent universality except that, this time, common refers to an uncountable family of densities. The following proposition strengthensone of the main results of [21] and gives an almost positive answer to Question 4.5. Proposition 4.6. We denote by D the set of all sequences α (with P k ≥ α k = + ∞ ) suchthat α . D s for some s ∈ N . If T ∈ L ( X ) satisfies the Frequent Hypercyclicity Criterion,then \ α ∈D FHC α ( T ) = ∅ . Proof. It is clearly enough to prove that T s ∈ N F HC D s ( T ) is non-empty. The proof is basedon the calculations led in [21, Section 3]. Let us consider the function f : N → N defined by f ( j ) = m for all j ∈ { a m , . . . , a m +1 − } with a m = 2 ... m where appears m times . Then we define the sequence ( n k ( f )) k ≥ as follows: n ( f ) = 2 and n k ( f ) = 2 k − X i =1 f ( δ i ) + f ( δ k ) for k ≥ , where δ j is the index of the first zero in the dyadic representation of j (for e.g., if k = 11 =1 . + 1 . + 0 . + 1 . , then δ k = 3 ). Lemma 3.8 of [21] ensures that for all s ≥ thereexist C , C , C > such that for all integer k large enough C k − C log ( s ) ( k ) ≤ n k ( f ) ≤ C k + C log ( s ) ( k ) . A similar calculation as that of [20, Lemma 4.10] allows to conclude that for all s ≥ d D s (( n k ( f ))) > . Therefore this sequence ( n k ( f )) allows to construct a hypercyclic vectorfor T which will be D s -frequently hypercyclic for all s ≥ (we refer the reader to thebeginning of Section 2 of [21]). (cid:3) References [1] E. Abakumov, J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal. , (2003), 494–504.[2] F. Bayart, Common hypercyclic vectors for composition operators, J. Operator Theory , (2004), no.2, 353–370.[3] F. Bayart, Common hypercyclic vectors for high-dimensional families of operators, Int. Math. Res. Not.IMRN Trans. Amer. Math. Soc. , (2006), no. 11,5083–5117.[5] F. Bayart, S. Grivaux, R. 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Joliot Curie, 13453 Marseille Cedex 13, France E-mail address : [email protected] Romuald Ernst, Univ. Littoral Côte d’Opale, UR 2597, LMPA, Laboratoire de Mathé-matiques Pures et Appliquées Joseph Liouville, Centre Universitaire de la Mi-Voix, Maisonde la Recherche Blaise-Pascal, 50 rue Ferdinand Buisson, BP 699, 62228 Calais, France E-mail address : [email protected] Monia Mestiri, Département de Mathématique, Institut Complexys, Université de Mons,20 Place du Parc, 7000 Mons, Belgium E-mail address : [email protected] Augustin Mouze, Laboratoire Paul Painlevé, UMR 8524, Cité Scientifique, 59650 Vil-leneuve d’Ascq, France, Current address: École Centrale de Lille, Cité Scientifique,CS20048, 59651 Villeneuve d’Ascq cedex, France E-mail address ::