Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

Given an affine symbol φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} and a multiplier w, we focus on the weighted composition operator Cw,φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{w, \varphi }$$\end{document} acting on the spaces Exp and Exp0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Exp^0$$\end{document} of entire functions of exponential and of infraexponential type, respectively. We characterize the continuity of the operator and, for w the product of a polynomial by an exponential function, we completely characterize power boundedness and (uniform) mean ergodicity. In the case of multiples of composition operators, we also obtain the spectrum and characterize hypercyclicity.


Introduction and Outline of the Paper
The purpose of this paper was to study the dynamics of the weighted composition operator C w,ϕ : f → w(f • ϕ) on the space Exp of entire functions of exponential type and on the space Exp 0 of entire functions of infraexponential type. The first space is formed by the entire functions which are of exponential type α for some α > 0, endowed with its natural locally convex topology which makes it an (LB)-space, and the second one by all the entire functions which are of exponential type for each α > 0, endowed with its natural locally convex topology which makes it a Fréchet space. Here we continue the research of the first author in [10], where the dynamics of the operator is studied on weighted Banach spaces of entire functions H v α , H 0 v α , defined by weights of exponential type v α (z) = e −αz , α > 0, z ∈ C. We refer to the next section for the precise notation and definitions.
In Sect. 3 we characterize the continuity of the operator when the symbol ϕ is an affine function, that is, when ϕ(z) = az + b, a, b ∈ C, and we show MJOM it is never compact. In the setting of the Banach spaces H v α and H 0 v α , the operator C w,ϕ is not continuous if |a| > 1, or if |a| = 1 and the multiplier w is not constant [10,Theorem 8]. On the spaces Exp and Exp 0 , for every a ∈ C we obtain continuity for multipliers of the form w(z) = p(z)e βz , β ∈ C, in the case of Exp and w(z) = p(z) in the case of Exp 0 , p being a polynomial. These weighted composition operators are natural in the following sense: if ϕ is an entire function and w(z) = p(z)e βz (resp. w(z) = p(z)), β ∈ C, p a polynomial, then C w,ϕ is continuous in Exp (resp. in Exp 0 ) if and only if ϕ(z) = az + b, a, b ∈ C. Moreover, if ϕ(z) = az + b, a, b ∈ C and w(z) = p(z)e q(z) , p, q being polynomials, then C w,ϕ is continuous in Exp (resp. in Exp 0 ) if and only if q(z) = βz, β ∈ C (resp. q ≡ 0).
The most relevant results we present are given in Sect. 4. Since the pioneer work of Bonet and Domański [18], the study of ergodic properties of composition operators and weighted composition operators in Banach and Fréchet spaces of analytic functions has become a very active area of research in mathematical analysis, see [7,8,10,12,13,23,28]. For natural weighted composition operators we completely characterize when C w,ϕ is power bounded and (uniformly) mean ergodic on Exp and on Exp 0 . Here, contrary to what happens in the Banach space H 0 v α , α > 0 [10, Theorem 16 b)], power boundedness is equivalent to (uniform) mean ergodicity.
Theorem ME-Exp. Let ϕ(z) = az + b, a, b ∈ C and w(z) = p(z)e βz , p being a polynomial and β ∈ C. The operator C w,ϕ is (uniformly) mean ergodic on Exp if and only if it is power bounded if and only if one of the following conditions occurs: (i) |a| < 1 and w b 1−a ≤ 1.
Theorem ME-Exp 0 . Let ϕ(z) = az + b, a, b ∈ C and let w(z) be a polynomial. The operator C w,ϕ is mean ergodic on Exp 0 if and only if it is power bounded and if and only if one of the following conditions occur: From our results it follows that C w,ϕ is (uniformly) mean ergodic on Exp (Exp 0 ) if and only if there exists α 0 > 0 such that C w,ϕ : is power bounded and mean ergodic for each α > α 0 (α > 0) except in the case of Theorem ME-Exp (ii), where if β = 0, there is no α > 0 such that [10,Theorem 8]). We point out that the theorems in Sect. 4 which are valid for symbols ϕ(z) = az + b, a = 1, are stated only for ϕ(z) = az, since the general case follows immediately from this reduction.
In Sect. 5 we focus our study on the case when w is a constant function, that is, on multiples of composition operators. In this case we give a complete description of the spectrum, and also completely characterize the hyperciclicity of the operator on Exp and on Exp 0 . More precisely, we get that if ϕ = az + b with a = 1, then λC ϕ cannot be hypercyclic (even weakly supercyclic). In fact, this is satisfied for general weighted composition operators. For multiples of the translation operator we get the following characterization: (i) It is hypercyclic, topologically mixing and chaotic on Exp for every λ ∈ C. (ii) It is topologically transitive on Exp 0 if and only if |λ| = 1. In this case, it is hypercyclic and topologically mixing.
Finally, we include an appendix where we improve some results of [10] for weighted composition operators defined on Banach spaces.

Notation and Preliminaries
Our notation is standard. We denote by H(C) the space of entire functions endowed with the compact open topology τ co of uniform convergence on the compact subsets of C, and by D the open unit disc centered at zero. Given two entire functions w and ϕ, the weighted composition operator C w,ϕ on H(C) is defined by The function ϕ is called symbol and w is called multiplier. C w,ϕ combines the composition operator We say that v : C →]0, ∞[ is a weight if it is continuous, decreasing and radial, that is, v(z) = v(|z|) for every z ∈ C. It is rapidly decreasing if lim For an arbitrary weight v on C, the weighted Banach spaces of entire functions with O-and o-growth conditions are defined as , τ co ) with continuous inclusions. If we assume v is rapidly decreasing, then H 0 v and H v contain the polynomials. We denote by B v and B 0 v the closed unit balls of H v and H 0 v , respectively. B v is compact with respect to τ co . Given the exponential weight v(z) = e −|z| , z ∈ C, consider the decreasing sequence (v n ) n and the increasing sequence of weights (v 1/n ) n . The The space Exp is an inductive limit of Banach spaces, that is, the increasing union of the spaces H v n with the strongest locally convex topology for which all the injections H v n → Exp, n ∈ N, become continuous. It is a (DFN)algebra [24]. The space Exp 0 is a projective limit of Banach spaces, that is, the decreasing intersection of the spaces H v 1/n , n ∈ N, whose topology is defined by the sequence of norms · v 1/n . It is a nuclear Fréchet algebra [25]. Clearly Exp 0 ⊂ Exp and the polynomials are contained and dense in both spaces, so Exp 0 is dense in Exp. The inductive limit Exp is boundedly retractive, that is, for every bounded subset B of Exp, there exists n ∈ N such that B is bounded in H 0 v n and the topologies of Exp and H 0 v n coincide on B. This is a stronger condition than being regular, i.e. for every bounded subset B of Exp, there exists n ∈ N such that B is bounded in H 0 v n . Weighted algebras of entire functions have been considered by many authors; see, e.g. [14,16,24,25,29] and the references therein.
Our notation for locally convex spaces and functional analysis is standard [26]. For a locally convex space E, cs(E) denotes a system of continuous seminorms determining the topology of E and the space of all continuous linear operators on E is denoted by L(E). Given T ∈ L(E) we say that x 0 ∈ E is a fixed point of T if T (x 0 ) = x 0 , and that it is periodic if there exists n ∈ N such that T n (x 0 ) = x 0 , where T n := T • n) · · · • T. A continuous linear operator T from a locally convex space E into itself is called hypercyclic if there is a vector x (which is called a hypercyclic vector) in E such that its orbit (x, T x, T 2 x, . . . ) is dense in E. Every hypercyclic operator T on E is topologically transitive in the sense of dynamical systems, that is, for every pair of non-empty open subsets U and V of E there is n such that T n (U ) meets V. The operator T is topologically mixing if for every pair of non-empty open subsets U and V of E there is n 0 such that for each n ≥ n 0 , T n (U ) meets V. T is chaotic if it is topologically transitive and has a dense set of periodic points.
An operator T ∈ L(E) is said to be power bounded if (T n ) n is an equicontinuous set of L(E). The spaces we are considering are barrelled; therefore, an operator T ∈ L(E) is power bounded if and only if, for each spaces, according to [2, Proposition 2.4], they are uniformly mean ergodic, that is, each power bounded operator on them is automatically uniformly mean ergodic. Clearly, if T is mean ergodic, then (T n x/n) n converges to 0 for each x ∈ E, and if it is uniformly mean ergodic, (T n /n) n converges to 0 in τ b . For a good exposition of ergodic theory we refer the reader to the monograph [27], and for the subject of linear dynamics, to the monographs by Bayart and Matheron [9] and by Grosse-Erdmann and Peris [21]. For If we need to stress the space E, then we write σ(T, E) and σ p (T, E).

Continuity and Compactness
An operator T : E → E on a locally convex space E is said to be compact (resp. bounded) if there exists a 0-neighbourhood U in E such that T (U ) is a relatively compact (resp. bounded) subset of E. Every bounded operator is continuous. If the bounded subsets of E are relatively compact, as it happens in Exp and Exp 0 , then bounded and compact operators coincide. T is said to be Montel if it maps bounded sets into relatively compact sets. So, every continuous operator on Exp and on Exp 0 is Montel. If E is a Banach space, T is bounded (resp. Montel) if and only if it is continuous (resp. compact). This is not satisfied on the spaces under consideration.
In order to study the continuity and compactness, first we need the following lemmata for inductive and projective limits of Banach spaces, respectively: Lemma 2 [5, Lemma 25]. Let E := proj m E m and F := proj n F n be Fréchet We are interested in the study of the dynamics of C w,ϕ when the multiplier is of the type w(z) = p N (z)e βz , p N a polynomial of degree N , and β ∈ C. The next result (see [10,Proposition 5] and [19, Proposition 3.1]) yields that for such multipliers, in order to have the continuity of C w,ϕ we must reduce to affine symbols, i.e. symbols of the form ϕ(z) = az +b for some a, b ∈ C. Proposition 3. Consider ϕ and w entire functions such that C w,ϕ : then ϕ must be affine. In particular, this happens when considering the multiplier w(z) = p N (z)e βz , β ∈ C in the case of Exp, and Accordingly, in the rest of the paper we only consider affine symbols. In the next proposition we obtain that, unlike what happens when the operator acts on the Banach space H v α , α > 0, the operator can be continuous on Exp and Exp 0 if |a| > 1 or if |a| = 1 and the multiplier w is not constant (see [10,Theorem 8]).
Proof. As C w,ϕ (1) = w, it is trivial that w must belong to the corresponding space if the operator is well defined. Let us see the converse. Assume first w ∈ Exp. Then there exists s ∈ N such that w ∈ H 0 v s . Observe that given n ∈ N, we can find m ∈ N, m > n|a| + s such that The continuity follows now by Lemma 1(i) and [10, Lemma 1]. By [10, Lemma 3], we also have that C w,ϕ : The continuity follows now by Lemma 2(i) and [10, Lemma 1]. By [10, Lemma 3], we get that C w,ϕ :

Proposition 5.
For ϕ(z) = az + b, a, b ∈ C, w = 0, the operator C w,ϕ is never compact on Exp and on Exp 0 . Thus, taking n > m/|a| we get that the entire function w converges to 0 as |z| → ∞, a contradiction if w ≡ 0. The case C w,ϕ : Exp 0 → Exp 0 is analogous using Lemma 2(ii) and [10, In the next result we characterize the continuity of the operator for a type of multipliers.
Proof. By Proposition 4, C w,ϕ will be continuous if and only if w belongs to the corresponding space. Given For z ∈ C and α > 0 we have So, if M ≤ 1 there exists n ∈ N such that w ∈ H v n ⊆ Exp. If M = 0, w ∈ H v 1/n for every n ∈ N, i.e. w ∈ Exp 0 , and the continuity holds. Conversely, if M ≥ 2 observe that w / ∈ H v α for each α > 0, hence the operator can not be continuous on Exp neither on Exp 0 . In the case of Exp 0 , if q M (z) = b 1 z, b 1 = 0, we can find c ∈ C, |c| = 1, and n ∈ N such that Thus, as w / ∈ H v 1/n the operator C w,ϕ can not be continuous on Exp 0 .

Power Boundedness and Mean Ergodicity
In this section, consider w ∈ Exp (resp. w ∈ Exp 0 ) and ϕ(z) = az + b, a, b ∈ C. We have seen that C w,ϕ : Exp → Exp (resp. C w,ϕ : Exp 0 → Exp 0 ) is continuous. Its iterates have the following expression: Observe that the symbol ϕ has a fixed point z 0 = b 1−a if and only if a = 1, and for k ∈ N, we get

MJOM
If we take f ≡ 1, we get a necessary condition for the operator to be power bounded or mean ergodic. Recall that Exp is boundedly retractive, i.e. each convergent sequence is convergent in some Banach space of the inductive limit.
In the next result we prove that the necessary condition for power boundedness in Proposition 7 is also sufficient when C ϕ is power bounded on each H v α , α > 0. This situation occurs for |a| ≤ 1, a = 1 (see [10,Theorem 22]), and obviously when a = 1 and b = 0. However, in general it is not sufficient. For instance, in Corollary 21 we obtain that the composition operator C ϕ is never mean ergodic on Exp and on Exp 0 if |a| > 1 or if a = 1 and b = 0. Proposition 8. Let |a| ≤ 1, a = 1 or a = 1, b = 0. The following is satisfied: Proof. Under the hypothesis we are considering, C ϕ : H v α → H v α is power bounded for every α > 0 (see [10,Theorem 22] for the non trivial case |a| ≤ 1, a = 1, b = 0). (i) Assume there exist m ∈ N and C > 0 such that for every k ∈ N, so the power boundedness holds. The converse follows by Proposition 7. For (ii) proceed analogously in order to get that for every f ∈ Exp 0 and every n ∈ N there exists D > 0 such that for every k ∈ N.
Since each non constant entire function w is unbounded, it is satisfied that there exists a ∈ C such that the sequence |w(a)| k k k is unbounded. As an immediate consequence, the next result on multiplication operators is satisfied as follows:  In the rest of the paper, for a = 1 we can assume without loss of generality that ϕ(z) = az. Indeed, in the next lemma we prove that for a = 1, every weighted composition operator is conjugated to another one with symbol ϕ(z) = az. So, the study of power boundedness and (uniform) mean ergodicity can be reduced to this case.
1−a and ϕ a (z) = az, z ∈ C, are conjugated. Proof. It is easy to see that the conjugacy holds through the homeomorphism So, the operator can be mean ergodic neither on Exp, nor on Exp 0 .
We have seen above that a necessary condition for the power boundedness and mean ergodicity of C w,ϕ is that the products of the iterates of the composition operator applied on the multiplier are bounded. In what follows, we consider multipliers of the form w(z) = p N (z)e βz , N ∈ N 0 , β ∈ C when considering the space Exp and w(z) = p N (z), N ∈ N 0 , when considering

ϕ is power bounded and (uniformly) mean ergodic if and only if
Proof. If |w(0)| > 1, the operator cannot be mean ergodic by Proposition 11(i).
If |w(0)| ≤ 1, [13, Theorem 3.10(i)] yields that for r > 0, there exists C > 1 such that, for every |z| ≤ r and k ∈ N, then k j=0 |w(a j z)| < C. On the other hand, observe that there exists M > 0 such that |p N (z)| ≤ M |z| N for every |z| ≥ r. Fix z 0 ∈ C and take j 0 ∈ N such that |a j z 0 | ≤ r for every j ∈ N, j ≥ j 0 . Therefore, for k ≥ j 0 we have Consider α > |β| 1−|a| and k ∈ N. For s = min(j 0 , k), we get the following: |w(a j z 0 )|e −α|z0| ≤ D for every k ∈ N. So, the conclusion holds by Proposition 8.
Proof. (i) Observe that there exist R > 0 and M > 0 such that |p N (z)| > M |z| N for every |z| > R. If N = 0 and k ∈ N, then By Lemma 12, if k is big enough, then sup r>R r Nk e −αr = sup r>0 r Nk e −αr .

Case a = 1
(i) If w(z) = p N (z)e βz , β ∈ C, N ∈ N 0 , then the operator C w,ϕ is not mean ergodic and thus not power bounded on Exp. N ∈ N 0 , the operator C w,ϕ is power bounded and mean ergodic on Exp 0 if and only if w ≡ λ, |λ| < 1.
Proof. If β = 0, given α > 0 we can find k ∈ N such that |β|k > α and θ k ∈ R such that the supremum in (4.2) is bigger than Therefore, w [k] / ∈ H v α and the operator is not mean ergodic and not power bounded on Exp by Proposition 7. If β = 0 and N = 0, there exist n 0 and C > 1 such that |p N (jb)| > C for j > n 0 . Now, the conclusion holds by Proposition 11(ii).
Finally, let us study the case w ≡ λ, λ ∈ C. For λC ϕ : Exp → Exp, take c ∈ C such that cb = |c||b| and |λ| > e −|cb| . The function e cz ∈ Exp and it satisfies (λC ϕ ) k e cz = (λe cb ) k e cz . So, as |λe cb | > 1, we cannot find n ∈ N such that (λCϕ) k (e cz ) k is bounded on H v n ; therefore, the operator is not mean ergodic on Exp.
In the case of λC ϕ : Exp 0 → Exp 0 , the operator is neither power bounded, nor mean ergodic if |λ| > 1 by Proposition 7(ii). If |λ| = 1, take f (z) = z ∈ Exp 0 in order to see that, evaluating at z = 0, Therefore, the operator cannot be mean ergodic, neither power bounded. If |λ| < 1, there exists n 0 ∈ N such that |λ| < e − |b| n for every n ≥ n 0 . [10, Theorem 19(i)] implies the operator λC ϕ : H v 1/n → H v 1/n is power bounded for every n ≥ n 0 , thus it is power bounded and uniformly mean ergodic on Exp 0 .

Multiples of Composition Operators
In this section we focus on multiples of composition operators, that is, operators of the form λC ϕ , λ ∈ C, ϕ(z) = az + b, a, b ∈ C. A complete characterization of power boundedness and mean ergodicity of these operators acting on weighted Banach spaces of entire functions can be found in [10], where hypercyclicity and the spectrum are also studied. Here we consider the study on the spaces Exp and Exp 0 .

Lemma 19.
The entire function f (z) = ∞ j=0 a j z j , a j , z ∈ C, satisfies the following: Proof. (i) Consider a = 1 and C ϕ : Exp → Exp. It is easy to see that the function e αz is an eigenvector associated with the eigenvalue e αb for every α ∈ C. Thus, C\{0} ⊆ σ p (C ϕ , Exp) ⊆ σ(C ϕ , Exp) and the conclusion is satisfied by Proposition 18.
The assertion follows because 1 is an eigenvalue. Indeed, constant functions are fixed points of the operator. By Lemma 10, in (ii) and (iii) we can assume without loss of generality that b = 0. The point spectrum in these cases follows from the fact that, if a = 1, then C ϕ z j = a j z j , j ∈ N 0 , and the proof of [  (ii) By the inclusions above, if a = 1 is a root of unity the conclusion is trivial, and if |a| < 1, the assertion holds because 0 / ∈ σ(C ϕ ) by Proposition 18. If |a| > 1, as σ(C ϕ ) = (σ(C ϕ −1 )) −1 with ϕ −1 (z) = 1 a z, the previous case yields the following conclusion. (iii) Assume |a| = 1 and a is not a root of unity. We prove that μ ∈ ρ(C ϕ ) if and only if there exists k > 0 and ε > 0 such that |a j − μ| ≥ εk −j for each j ∈ N. Let us analyse first σ(C ϕ , Exp). We have seen above that, if the condition is satisfied, C ϕ − μI is injective. We prove it is also surjective. Given g(z) = ∞ j=0 a j z j ∈ Exp, by Lemma 19(i) there exists B, C > 0 such that |a j | ≤ C B j j! for every j ∈ N. It is easy to see that the function f (z) := ∞ j=0 aj a j −μ z j , z ∈ C, satisfies (C ϕ − μI)f = g and that there exists D > 0 such that Again by Lemma 19, we get f ∈ Exp, and so, μ ∈ ρ(C ϕ , Exp). For the converse, assume μ ∈ ρ(C ϕ , Exp). Then (C ϕ − μI) −1 : Exp → Exp exists, it is continuous and satisfies The continuity implies (see Lemma 1) that for n = 1 there exist k ∈ N and C > 0 such that, for each monomial z j ∈ Exp, j ∈ N 0 , we have By Lemma 12, for every j ∈ N 0 we get j ke i.e. |a j − μ| ≥ ε k j for some ε > 0.

Dynamics
As an immediate consequence of the theorems in Sect. 4, we get the following: MJOM licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.
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Appendix
We finish the paper with two results on mean ergodicity on weighted Banach spaces of entire functions which improve some parts of [10,Theorem 16]. The first one must be compared with [10, Theorem 16 a)].
Proof. Without loss of generality, we restrict to the case b = 0 (see Lemma 10). By [10,Theorem 8] and proceeding as in the proof of Theorem 13, we get the operator C w,ϕ is power bounded and compact on H v α and on H 0 v α for every α > |β| 1−|a| . As a consequence, C w,ϕ is uniformly mean ergodic on these spaces by Yosida-Kakutani Mean Ergodic Theorem [30]. Proof. By [10,Theorem 16(b)], it only remains to show that for a ∈ D\[0, 1), the operator is mean ergodic. Without loss of generality, assume b = 0. Observe that for every f ∈ H 0 v α and every z ∈ C, C k w,ϕ f (z) = e βz 1−a k 1−a f (a k z) ∈ (0, 1)). Proceeding as in the proof of [23, Lemma 3.4(1)], we get C k w,ϕ f converges to e βz 1−a δ 0 (f ) as k tends to infinity in the weak topology. As the operator is power bounded, the operator is mean ergodicity by Yosida's theorem [27,Theorem 1.3,p.26].