Frequently hypercyclic translation semigroups

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized.


Introduction and preliminaries
A continuous linear operator T on a separable Banach space is called hypercyclic if there is an element x ∈ X, called hypercyclic vector, such that the orbit {T n x : n ∈ N} is dense in X. The first historically known examples of hypercyclic operators are due to Birkhoff, MacLane and Rolewicz. In particular, the last author studied hypercyclicity in the setting of weighted shift operators on l p and c 0 . The interest in the study of linear dynamics of shift operators is nowadays still alive, since many classical operators (e.g. derivative operator in spaces of entire functions) can be viewed as such operators. We refer to the recent monographs [8] and [24] for a complete overview on the subject. This notion has been deeply investigated by various authors, see e.g. [23,14,18]. In particular frequently hypercyclic weighted shifts have been investigated in [14,11], until their behaviour has been completely characterized in l p and c 0 by Bayart and Rusza [9]. In parallel with the theory for linear operators, since the seminal paper by Desch, Schappacher and Webb [20], many researchers turned their attention to the hypercyclic behaviour of strongly continuous semigroups. Actually hypercyclicity appears in solution semigroups of evolution problems associated with "birth and death" equations for cell populations, transport equations, first order partial differential equations, Black and Scholes equations, diffusion operators as Ornstein-Uhlenbeck operators [2,4,5,6,12,13,15,17,21,25,27,28].
We recall that, if X be a separable infinite-dimensional Banach space, a C 0 -semigroup (T t ) t≥0 of linear and continuous operators on X is said to be hypercyclic if there exists x ∈ X (called hypercyclic vector for the semigroup) such that the set {T t x : t ≥ 0} is dense in X. An element x ∈ X is said to be a periodic point for the semigroup if there exists t > 0 such that T t x = x. A semigroup (T t ) t≥0 is called chaotic if it is hypercyclic and the set of periodic points is dense in X.
The role of "test" class, which is played by weighted shifts in the setting of discrete linear dynamical systems, is covered by translation semigroups in the setting of continuous linear dynamical systems.
An admissible weight function on R is a strictly positive measurable function ρ : R → R for which there exist constants M ≥ 1 and ω ∈ R such that ρ(τ ) ≤ Me ωt ρ(τ + t) for all τ ∈ [0, +∞[ and all t > 0.
If ρ is an admissible weight, then for every l > 0 there exist A, B > 0 such that for every σ ∈ R and for every t ∈ [σ, σ + l], it holds Consider the following function spaces: If X is any of the spaces above and ρ is an admissible weight, the translation semigroup T = (T t ) t≥0 is defined as usual by and is a C 0 -semigroup (see e.g. [20]). If X is one of the spaces L ρ p (R) or C ρ 0 (R) with an admissible weight function ρ, the translation semigroup T on X is hypercyclic if and only if lim inf t→+±∞ ρ(t) = 0.
If X = C ρ 0 (R), then the translation semigroup T on X is chaotic if and only if lim x→±∞ ρ(x) = 0.
If X = L ρ p (R), T is chaotic if and only if k∈Z ρ(k) < ∞ [19,20,27,28]. The concept of frequent hyperyclicity was extended to C 0 -semigroups in [3]. The lower density of a measurable set M ⊂ R + is defined by where µ is the Lebesgue measure on R + .
A C 0 -semigroup (T t ) t≥0 is said to be frequently hypercyclic if there exists x ∈ X such that Dens({t ∈ R + : T t x ∈ U}) > 0 for any non-empty open set U ⊂ X. In [16,26], it was proved that x ∈ X is a (frequently) hypercyclic vector for (T t ) t≥0 if and only if x is a (frequently) hypercyclic vector for each single operator T t , t > 0. However, this is not the case in general if we consider the chaos property [12].
In [26], it was proved a continuous version of the Frequent Hypercyclicity criterion based on the Pettis integral and that chaotic translation semigroups on weighted spaces of integrable or continuous functions on the real line are frequently hypercyclic.
Moreover, in [29], it is proved that the Frequent Hypercyclicity criterion for semigroups implies the existence of strongly-mixing Borel probability measures with full support.
In this paper we characterize, in the line of [9], frequently hypercyclic translation semigroups on weighted function spaces. The main results are Theorems 5 and 11 that are proved in the last section. In particular, Theorem 5 will be consequence of Theorem 1, which characterizes frequent hypercyclicity of the so-called pseudo-shifts on spaces c 0 (I), where I is a countably infinite set.

Frequently hypercyclic weighted pseudo-shift
We recall the concept of weighted pseudo-shift that was introduced by Grosse-Erdmann in [22]. Given X, Y topological sequence spaces over countably infinite sets I and J, a continuous linear operator T : X → Y is called a weighted pseudo-shift if there is a sequence (b j ) j∈J of non-zero scalars and an injective mapping φ : We will be interested in weighted pseudo-shifts acting on spaces of vanishing sequences. More precisely, given a countable set I, we consider the space The first result that we prove is a characterization of frequently universal sequences of weighted pseudo-shifts on c 0 (I).
We recall that a sequence (T n ) n∈N of continuous mappings between topological spaces X and Y is called frequently universal if there exists x ∈ X such that for every non-empty open set U ⊆ Y , dens{n ∈ N : T n x ∈ U} > 0.
Following the idea of Bayart and Ruzsa in [9] for weighted backward shifts on c 0 (Z), we first obtain a characterization for weighted pseudo-shifts.
then |n − m| ≤ |g(s) − g(t)| and that (φ n ) n is a run-away sequence, i.e. for each pair of finite subsets I 0 , J 0 ⊂ I there exists an n 0 ∈ N such that, for every n ≥ n 0 , φ n (J 0 )∩I 0 = ∅. Then (T n ) n is frequently universal on c 0 (I) if and only if there exist ( for all) a sequence (M(p)) p∈N of positive real numbers tending to ∞, a sequence (E p ) p∈N of subsets of N and an increasing sequence (W p ) p∈N of finite subsets of I with I = ∞ p=1 W p , such that: (a) For any p ≥ 1, dens(E p ) > 0.
Proof. We first observe that we may replace "there exists a sequence (M(p))" by "for any sequence (M(p))" in the statemente of the theorem. Indeed, if properties (a) to (d) are true for some sequence (M(p)), then they are also satisfied for any subsequence of it. Consider a sequence (α p ) p∈N of positive real numbers such that α 1 = 2 and for all i∈I be a frequently universal vector for (T n ) n and set Clearly dens(E p ) > 0. Let p = q, n ∈ E p , m ∈ E q and let us show that φ n (W p )∩φ m (W q ) = ∅. Assume that p < q. By contradiction, let us assume that there exist s ∈ W p and t ∈ W q such that φ n (s) = φ m (t). The s-th coefficient of T n x is b n s x φn(s) , then But this contradicts the definition of (α p ). Now let n ∈ E p and s ∈ W p , the s-th coefficient of T n x is b n s x φn(s) and its modulus cannot be less than αp 2 . Let M > 0. Given ǫ = αp 2M , since x ∈ c 0 (I), there exists J ⊂ I finite such that |x i | < ǫ for all i ∈ I \ J. As φ n is a run-away sequence there exists n 0 ∈ N such that for all n ∈ E p , n > n 0 and for all s ∈ W p , φ n (s) / ∈ J, and then |x φn(s) | < ǫ. As a result, for all n ≥ n 0 and s ∈ W p : So, we have proved (c). Finally, let n ∈ E p , m ∈ E q , t ∈ W q such that φ n (s) = φ m (t), then s / This shows (d) with M(p) = p. We now show that the condition is sufficient. We may assume that, for any p ≥ 1, Clearly, E ′ p is a cofinite subset of E p due to (c), hence dens(E ′ p ) > 0. Let (y p ) p≥0 be a dense sequence in c 0 (I) such that supp(y p ) ⊂ W p and ||y p || < ρ p . We define x ∈ R I by setting This definition is not ambiguous because given n ∈ E ′ p , m ∈ E ′ q we have φ n (W p ) ∩ φ m (W q ) = ∅. If s 1 , s 2 ∈ W p , s 1 = s 2 , then φ n (s 1 ) = φ n (s 2 ) because φ n is injective. Moreover, if n, m ∈ E ′ p are such that φ n (s) = φ m (s), by hypothesis |n − m| ≤ |g(s) − g(s)| = 0, then n = m.
We claim that x ∈ c 0 (I). Indeed, given ǫ > 0, there exists p 0 ∈ N such that for p ≥ p 0 and n ∈ E ′ p , s ∈ W p : If p ≤ p 0 : We finally show that x is a frequently hypercyclic vector by proving that for all p ≥ 1, n ∈ E ′ p , ||T n x − y p || < ǫ(p) with ǫ(p) → 0 as p → ∞. Observe that ||T n x − y p || = sup s / ∈Wp |b n s x φn(s) |.
The terms which appear in the norm are nonzero if and only if φ n (s) = φ m (t), m ∈ E ′ q , t ∈ W q and for these terms it holds that As a corollary, we obtain a characterization of frequent hypercyclicity for weighted backward shifts operators defined on c 0 (I), in the case that I ⊆ R.
Corollary 2. Let I be a countably infinite subset of R such that I + Z ⊂ I, I = ∞ p=1 W p , where (W p ) p is an increasing sequence of finite subsets. Let (w i ) i∈I be a bounded and bounded below sequence of positive integers. The operator T : c 0 (I) → c 0 (I) defined by T (x i ) i∈I = (w i+1 x i+1 ) i∈I is frequently hypercyclic on c 0 (I) if and only if there exist ( for all) a sequence (M(p)) of positive real numbers tending to ∞ and a sequence (E p ) of subsets of N such that (a) For any p ≥ 1, dens(E p ) > 0.
(d) For any p, q ≥ 1, n ∈ E p , m ∈ E q , n = m and t ∈ W q : Proof. This corollary is a particular case of Theorem 1 when we consider T n = T n with Remark 3. Observe that condition (d) is equivalent to say that for any p, q ≥ 1, n ∈ E p , m ∈ E q , n = m and t ∈ W q : and we obtain the conditions of Theorems 12 in [9].

Frequently hypercyclic translation semigroups
The purpose of this section is to obtain a characterization of frequent hypercyclicity for translation semigroups on C ρ 0 (R) and L ρ p (R). To treat the case of continuous functions, we will first need to recall some known results about the construction of a Schauder basis in C 0 (R), referring for more details to [30].
Set I = Z + D and consider the partition I = n≥0 V n where V 0 = {0, 1}, and We define an order on I assuming that the elements of V k are earlier than the elements of V n if 0 ≤ k < n, and within each V n we keep the usual order.
The system (ϕ i ) i∈I , is a Schauder basis in C 0 (R). More precisely, if f ∈ C 0 (R), then f = k+τ ∈Z+ D a k+τ φ k+τ where Lemma 4. Let ρ be an admissible weight function on R such that ρ(x) = ρ([x]) for any x ∈ R and let T 1 : C ρ 0 (R) → C ρ 0 (R) be the translation operator defined as T 1 f (x) = f (x + 1). Then T 1 is quasi conjugated to the weighted backward shift operator B w : Proof.
Theorem 5. Let T be the translation semigroup on C ρ 0 (R), where ρ is an admissible function and sup k∈Z ρ(k+1) ρ(k) < ∞. T is frequently hypercyclic on C ρ 0 (R) if and only if there exist a sequence (M(p)) of positive real numbers tending to ∞ and a sequence (E p ) of subsets of N such that: (a) For any p ≥ 1, dens(E p ) > 0.
(d) For any p, q ≥ 1, for any n ∈ E p and any m ∈ E q , n = m and for all k ∈ [−q, q + 1] : where Proof. Let us point out that if ρ is an admissible weight function then sup ρ(k) ρ(k+1) < ∞. By hypothesis we have sup ρ(k+1) ρ(k) = M < ∞, then there exist constants 0 < A < B such that .
We conclude the result combining Corollary 2 and Lemma 4.
Remark 6. Let (E p ) be a sequence of subsets of N such that for any p, q ≥ 1, p = q, we have that τ − σ ∈ Z. Thus straightforward calculations give that h = k and |u − v| = a2 h−1 with a ∈ Z + . On the other hand, |u − v| < 2 h−1 , hence a = 0. Therefore τ = σ and so n + s = m + t. Now the assertion follows by the properties of the set E p .
As an immediate consequence, we get that if ρ be an admissible weight function on R such that sup k∈Z ρ(k+1) ρ(k) < ∞ and we set w k = ρ(k) ρ(k+1) , k ∈ Z, then, by [9][Theorem 9] and by Theorem 7, if B w is frequently hypercyclic on c 0 (Z), also the translation semigroup is frequently hypercyclic on C ρ 0 (R).
Proposition 7. Let T be a mixing (equivalently chaotic) translation C 0 -semigroup on C ρ 0 (R). Then T is frequently hypercyclic.
Proof. As it is proved in [13,27,28], chaos and mixing are equivalent properties for the translation C 0 -semigroup on C ρ 0 (R), and this happens if and only if lim x→±∞ ρ(x) = 0. Consider a sequence (E p ) of subsets of N such that for any p ≥ 1, dens(E p ) > 0 and for any p, q ≥ 1, p = q, (E p + [−p, p]) ∩ (E q + [−q, q]) = ∅. (see e.g. the constructions in [8]). Assumption (c) of Theorem 5 is clearly verified, while (b) is satisfied by Remark 6. Moreover, given n ∈ E p , m ∈ E q and k ∈ [−q, q + 1], we can define for each k ∈ [−q, q + 1], i ∈ N: It is clear that for n ∈ E p , m ∈ E q , |m − n| ≥ max(p, q), and Hence and hypothesis (d) is satisfied by the sequence M ′ (p) = M(p) and therefore T is frequently hypercyclic.
The final part of the paper will be devoted to the proof of Theorem 2. To this end we establish a relation between the discrete and the continuous case. We recall that the relation between the discrete and continuous case for Devaney chaos was studied in [12] and for distributional chaos in [7].
The following lemma follows immediately by the conjugacy of the backward shift on ℓ v p = {(x k ) k∈Z : k∈Z |x k |v k < ∞} and the weighted backward shift B w on ℓ p where w k = v k−1 v k and therefore dens{n ∈ N : ||B n x − y|| < ǫ} > 0 because f is a frequently hyperciclic vector. We have: By the density of finite sequences in ℓ v p we get that B is frequently hypercyclic.
Finally we are able to characterize frequently hypercyclic translation semigroups in L ρ p (R).
Theorem 11. Let ρ be an admissible function on R. The following assertions are equivalent: (1) the translation semigroup T is frequently hypercyclic.