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

Abstract

Given an affine symbol $\varphi$ and a multiplier w,  we focus on the weighted composition operator $C_{w, \varphi }$ acting on the spaces Exp and $Exp^0$ 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.

1 Introduction and Outline of the Paper

The purpose of this paper was to study the dynamics of the weighted composition operator $C_{w, \varphi }: f\rightarrow w(f\circ \varphi )$ 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 $\alpha$ for some $\alpha >0$, 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 $\alpha >0$, 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 $H_{v^{\alpha }},$ $H_{v^{\alpha }}^0,$ defined by weights of exponential type $v^{\alpha }(z)=e^{-\alpha z},$ $\alpha >0,$ $z\in {\mathbb {C}}.$ 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 $\varphi$ is an affine function, that is, when $\varphi (z)=az+b$, $a,b\in {\mathbb {C}},$ and we show it is never compact. In the setting of the Banach spaces $H_{v^{\alpha }}$ and $H_{v^{\alpha }}^0,$ the operator $C_{w,\varphi }$ 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 $a\in {\mathbb {C}}$ we obtain continuity for multipliers of the form $w(z)=p(z)e^{\beta z},$ $\beta \in {\mathbb {C}},$ 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 $\varphi$ is an entire function and $w(z)=p(z)e^{\beta z}$ (resp. $w(z)=p(z)$), $\beta \in {\mathbb {C}},$ p a polynomial, then $C_{w,\varphi }$ is continuous in Exp (resp. in $Exp^0$) if and only if $\varphi (z)=az+b$, $a,b\in {\mathbb {C}}$. Moreover, if $\varphi (z)=az+b$, $a,b\in {\mathbb {C}}$ and $w(z)=p(z)e^{q(z)}$, $p, \ q$ being polynomials, then $C_{w,\varphi }$ is continuous in Exp (resp. in $Exp^0$) if and only if $q(z)=\beta z,$ $\beta \in {\mathbb {C}}$ (resp. $q\equiv 0$).

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 $C_{w, \varphi }$ is power bounded and (uniformly) mean ergodic on Exp and on $Exp^0$. Here, contrary to what happens in the Banach space $H_{v^\alpha }^0,$ $\alpha >0$ [10, Theorem 16 b)], power boundedness is equivalent to (uniform) mean ergodicity.

Theorem ME-$\varvec{Exp}.$ Let $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}$ and $w(z)=p(z)e^{\beta z},$ p being a polynomial and $\beta \in {\mathbb {C}}$. The operator $C_{w,\varphi }$ 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 $\left| w\left( \frac{b}{1-a}\right) \right| \le 1.$

• (ii) $|a|=1,$ $a\ne 1$, $w(z)=\lambda e^{\beta z}$, $|\lambda |\le \left| e^{-\beta \frac{b}{1-a}}\right| ,$ $\beta \in {\mathbb {C}}.$

• (iii) $a=1$, $b=0$, $w(z)\equiv \lambda$, $|\lambda |\le 1$ (multiplication operator case).

Theorem ME-$\varvec{Exp^0}.$ Let $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}$ and let w(z) be a polynomial. The operator $C_{w,\varphi }$ 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:

• (i) $|a|<1$ and $\left| w\left( \frac{b}{1-a}\right) \right| \le 1.$

• (ii) $|a|=1,$ $a\ne 1$, $w(z)\equiv \lambda$, $|\lambda |\le 1.$

• (iii) $a=1$, $b=0$, $w(z)\equiv \lambda$, $|\lambda |\le 1$ (multiplication operator case).

• (iv) $a=1$, $b\ne 0$, $w(z)\equiv \lambda$, $|\lambda |< 1$.

From our results it follows that $C_{w, \varphi }$ is (uniformly) mean ergodic on Exp ($Exp^0$) if and only if there exists $\alpha _0>0$ such that $C_{w, \varphi }:H_{v^\alpha }^0\rightarrow H_{v^\alpha }^0$ is power bounded and mean ergodic for each $\alpha >\alpha _0$ $(\alpha >0)$ except in the case of Theorem ME-Exp (ii), where if $\beta \ne 0,$ there is no $\alpha >0$ such that $C_{w, \varphi }( H_{v^\alpha }^0)\subseteq H_{v^\alpha }^0$, as w is not constant (see [10, Theorem 8]). We point out that the theorems in Sect. 4 which are valid for symbols $\varphi (z)=az+b$, $a\ne 1,$ are stated only for $\varphi (z)=az$, 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 $\varphi = az+b$ with $a\ne 1$, then $\lambda C_{\varphi }$ 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:

Theorem Hyp. The weighted translation operator $\lambda T_b: f(z)\rightarrow \lambda f(z+b),$ $\lambda , b \in {\mathbb {C}},$ $b\ne 0,$ satisfies:

• (i) It is hypercyclic, topologically mixing and chaotic on Exp for every $\lambda \in {\mathbb {C}}.$

• (ii) It is topologically transitive on $Exp^0$ if and only if $|\lambda |=1.$ 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.

2 Notation and Preliminaries

Our notation is standard. We denote by $H({\mathbb {C}})$ the space of entire functions endowed with the compact open topology $\tau _{co}$ of uniform convergence on the compact subsets of ${\mathbb {C}},$ and by ${\mathbb {D}}$ the open unit disc centered at zero. Given two entire functions w and $\varphi ,$ the weighted composition operator $C_{w,\varphi }$ on $H({\mathbb {C}})$ is defined by

$$\begin{aligned} C_{w, \varphi }(f)=w (f \circ \varphi ),\ f\in H({\mathbb {C}}). \end{aligned}$$

The function $\varphi$ is called symbol and w is called multiplier. $C_{w,\varphi }$ combines the composition operator $C_{\varphi }: f \mapsto f \circ \varphi$ with the pointwise multiplication operator $M_{w}: f \mapsto w \cdot f$.

We say that $v:{\mathbb {C}}\rightarrow ]0, \infty [$ is a weight if it is continuous, decreasing and radial, that is, $v(z)=v(|z|)$ for every $z\in {\mathbb {C}}.$ It is rapidly decreasing if $\underset{r\rightarrow \infty }{\lim } r^kv(r)=0$ for all $k\in {\mathbb {N}}.$

For an arbitrary weight v on ${\mathbb {C}},$ the weighted Banach spaces of entire functions with O- and o-growth conditions are defined as

$$\begin{aligned} H_v= & {} \left\{ f \in {H}({\mathbb {C}}) : \Vert f\Vert _v:= \underset{z\in {\mathbb {C}}}{\sup }\,v(z)|f(z)| < \infty \right\} , \\ H_v^{0}= & {} \left\{ f \in {H}({\mathbb {C}}): \lim _{|z|\rightarrow \infty }v(z)|f(z)|=0 \right\} . \end{aligned}$$

$(H_v,\Vert \ \Vert _v)$ and $(H_v^0,\Vert \ \Vert _v)$ are Banach spaces, and $(H_v^0, \Vert \ \Vert _v) \hookrightarrow (H_v, \Vert \ \Vert _v)\hookrightarrow ({H}({\mathbb {C}}), \tau _{co})$ with continuous inclusions. If we assume v is rapidly decreasing, then $H_v^0$ and $H_v$ contain the polynomials. We denote by $B_v$ and $B_v^0$ the closed unit balls of $H_v$ and $H_v^{0}$, respectively. $B_v$ is compact with respect to $\tau _{co}$.

Given the exponential weight $v(z)=e^{-|z|},$ $z\in {\mathbb {C}},$ consider the decreasing sequence $(v^n)_n$ and the increasing sequence of weights $(v^{1/n})_n.$ The inclusions $H_{v^n}\hookrightarrow H_{v^{n+1}}^0$ and $H_{v^{1/(n+1)}}\hookrightarrow H_{v^{1/n}}^0,$ $n\in {\mathbb {N}},$ are compact. We are interested in the space of entire functions of exponential type

$$\begin{aligned} Exp:=ind_{n\in {\mathbb {N}}}H_{v^n}=ind_{n\in {\mathbb {N}}}H_{v^n}^0=\{f\in H({\mathbb {C}}):\ \exists n\in {\mathbb {N}}\text { such that } f\in H_{v^n}^0\} \end{aligned}$$

and of infraexponential type

$$\begin{aligned} Exp^0:= & {} proj_{n\in {\mathbb {N}}}H_{v^{1/n}}=proj_{n\in {\mathbb {N}}}H_{v^{1/n}}^0\\= & {} \{f\in H({\mathbb {C}}): f\in H_{v^{1/n}}^0 \quad \text { for every }n\in {\mathbb {N}}\}. \end{aligned}$$

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 $H_{v^n}\hookrightarrow Exp,$ $n\in {\mathbb {N}},$ 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 $H_{v^{1/n}},$ $n\in {\mathbb {N}},$ whose topology is defined by the sequence of norms $\Vert \cdot \Vert _{v^{1/n}}.$ It is a nuclear Fréchet algebra [25]. Clearly $Exp^0 \subset Exp$ 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 $n \in {\mathbb {N}}$ such that B is bounded in $H_{v^n}^0$ and the topologies of Exp and $H_{v^n}^0$ coincide on B. This is a stronger condition than being regular, i.e. for every bounded subset B of Exp, there exists $n \in {\mathbb {N}}$ such that B is bounded in $H_{v^n}^0$. 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, $\mathrm{cs}(E)$ 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 $T\in L(E)$ we say that $x_0\in E$ is a fixed point of T if $T(x_0)=x_0,$ and that it is periodic if there exists $n\in {\mathbb {N}}$ such that $T^n(x_0)=x_0,$ where $T^n:=T\circ \overset{n)}{\cdots }\circ T.$ 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 $(x,Tx,T^2x,\dots )$ 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 $n\ge n_0$, $T^n(U)$ meets V. T is chaotic if it is topologically transitive and has a dense set of periodic points.

An operator $T\in L(E)$ 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 $T\in L(E)$ is power bounded if and only if, for each $x\in E,$ its orbit $(x,Tx,T^2x,\dots )$ is bounded in E. T is called mean ergodic if the Cesàro means $(T_{[n]}x)_n,$ $T_{[n]}x:=\frac{1}{n}\sum _{j=1}^nT^jx, \, x\in E,$ converge in E. If the sequence of the Cesàro means of the iterates of T converges in L(E) endowed with the topology $\tau _b$ of uniform convergence on bounded sets, the operator T is called uniformly mean ergodic. Mean ergodic operators in barrelled locally convex spaces have been considered, e.g. in [1,2,3,4,5,6, 11]. As Exp and $Exp^0$ are Montel 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 $x\in E,$ and if it is uniformly mean ergodic, $(T^n/n)_n$ converges to 0 in $\tau _b.$

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 $T\in L(E),$ the resolvent set $\rho (T)$ of T consists of all $\lambda \in {\mathbb {C}}$ such that $(\lambda I-T)^{-1}$ exists in L(E). The set $\sigma (T):={\mathbb {C}}{\setminus } \rho (T)$ is called the spectrum of T. The point spectrum $\sigma _p(T)$ of T consists of all $\lambda \in {\mathbb {C}}$ such that $\lambda I-T$ is not injective. If we need to stress the space E,  then we write $\sigma (T, E)$ and $\sigma _p(T, E).$

3 Continuity and Compactness

An operator $T: E\rightarrow E$ 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 1

[6, Lemma 4.1]. Let $E=ind_nE_n$ and $F=ind_mF_m$ be two (LB)-spaces which are increasing unions of Banach spaces $E=\cup _n E_n$ and $F=\cup _m F_m.$ Let $T: E\rightarrow F$ be a linear map.

• (i) T is continuous if and only if for each $n\in {\mathbb {N}}$ there exists $m\in {\mathbb {N}}$ such that $T(E_n)\subseteq F_m$ and the restriction $T_{n,m}: E_n\rightarrow F_m$ is continuous.

• (ii) Assume that F is a regular (LB)-space. Then T is bounded if and only if there exists $m\in {\mathbb {N}}$ such that $T(E_n)\subseteq F_m$ and $T:E_n\rightarrow F_m$ is continuous for all $n\in {\mathbb {N}}.$

Lemma 2

[5, Lemma 25]. Let $E:= proj_mE_m$ and $F:=proj_nF_n$ be Fréchet spaces such that $E=\cap _mE_m$ with each $(E_m, \Vert \ \Vert _m)$ a Banach space and $F=\cap _nF_n$ with each $(F_n, \Vert \ \Vert _n)$ a Banach space. Moreover, assume E is dense in $E_m$ and that $E_{m+1}\subseteq E_m$ with a continuous inclusion for each $m\in {\mathbb {N}}$ (resp. $F_{n+1}\subseteq F_n$ with a continuous inclusion for each $n\in {\mathbb {N}}$). Let $T: E\rightarrow F$ be a linear operator.

• (i) T is continuous if and only if for each $n\in {\mathbb {N}}$ there exists $m \in {\mathbb {N}}$ such that T has a unique continuous linear extension $T_{m,n}: E_m\rightarrow F_n.$

• (ii) Assume that T is continuous. Then T is bounded if and only if there exists $m\in {\mathbb {N}}$ such that, for every $n\in {\mathbb {N}},$ the operator T has a unique continuous linear extension $T_{m,n}: E_{m}\rightarrow F_n.$

We are interested in the study of the dynamics of $C_{w, \varphi }$ when the multiplier is of the type $w(z)=p_N(z)e^{\beta z},$ $p_N$ a polynomial of degree N, and $\beta \in {\mathbb {C}}$. The next result (see [10, Proposition 5] and [19, Proposition 3.1]) yields that for such multipliers, in order to have the continuity of $C_{w, \varphi }$ we must reduce to affine symbols, i.e. symbols of the form $\varphi (z)=az+b$ for some $a, b\in {\mathbb {C}}.$

Proposition 3

Consider $\varphi$ and w entire functions such that $C_{w, \varphi }: H_{v^{\alpha }}\rightarrow H_{v^\gamma },$ $0<\alpha <\gamma ,$ is continuous. If there exists $\lambda >1$ such that $\lim _{|z|\rightarrow \infty }|w(z)|e^{|z|(\lambda \alpha -\gamma )}=\infty ,$ then $\varphi$ must be affine. In particular, this happens when considering the multiplier $w(z)=p_N(z)e^{\beta z},$ $\beta \in {\mathbb {C}}$ in the case of Exp,  and $w(z)=p_N(z)$ in the case of $Exp^0.$

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 $H_{v^{\alpha }},$ $\alpha >0,$ 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]).

Proposition 4

For $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}},$ the operator $C_{w, \varphi }$ is continuous on Exp (resp. $Exp^0$) if and only if $w\in Exp$ (resp. $w\in Exp^0$).

Proof

As $C_{w, \varphi }(1)=w,$ it is trivial that w must belong to the corresponding space if the operator is well defined. Let us see the converse. Assume first $w\in Exp.$ Then there exists $s\in {\mathbb {N}}$ such that $w\in H_{v^s}^0.$ Observe that given $n\in {\mathbb {N}},$ we can find $m\in {\mathbb {N}},$ $m>n|a|+s$ such that

$$\begin{aligned}&\lim _{|z|\rightarrow \infty }|w(z)|e^{n|az+b|-m|z|}\le \lim _{|z|\rightarrow \infty }|w(z)|e^{|z|(n|a|-m)}e^{n|b|}\\&\quad \le \lim _{|z|\rightarrow \infty }|w(z)|e^{-s|z|}e^{n|b|}= 0. \end{aligned}$$

The continuity follows now by Lemma 1(i) and [10, Lemma 1]. By [10, Lemma 3], we also have that $C_{w, \varphi }:H_{v_n}^0\rightarrow H_{v_m}^0$ is compact.

Assume now $w\in Exp^0.$ Given $n\in {\mathbb {N}}$ there exist $m\in {\mathbb {N}}$ and $s\in {\mathbb {N}}$ such that $0<\frac{1}{s}<\frac{1}{n}-\frac{1}{m}|a|.$ Therefore,

$$\begin{aligned}&\lim _{|z|\rightarrow \infty }|w(z)|e^{\frac{1}{m}|az+b|-\frac{1}{n}|z|}\le \lim _{|z|\rightarrow \infty }|w(z)|e^{-|z|(\frac{1}{n}-\frac{1}{m}|a|)}e^{\frac{|b|}{m}}\\&\quad \le \lim _{|z|\rightarrow \infty }|w(z)|e^{-\frac{1}{s}|z|}e^{\frac{|b|}{m}}=0. \end{aligned}$$

The continuity follows now by Lemma 2(i) and [10, Lemma 1]. By [10, Lemma 3], we get that $C_{w, \varphi }:H_{v^{1/m}}^0\rightarrow H_{v^{1/n}}^0$ is compact. $\square$

Proposition 5

For $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}},$ $w\ne 0,$ the operator $C_{w, \varphi }$ is never compact on Exp and on $Exp^0.$

Proof

Let us study first the case $C_{w, \varphi }: Exp\rightarrow Exp.$ Assume the operator is compact, i.e. bounded. By Lemma 1(ii) and [10, Lemma 1], there exists $m\in {\mathbb {N}}$ such that $\sup _{z\in {\mathbb {C}}}|w(z)|e^{n|az+b|-m|z|}<\infty$ for all $n\in {\mathbb {N}}.$ So, there exists $C_n>0$ such that, for every $z\in {\mathbb {C}},$

$$\begin{aligned} |w(z)|\le C_ne^{m|z|-n|az+b|}\le C_ne^{n|b|}e^{|z|(m-n|a|)}. \end{aligned}$$

Thus, taking $n>m/|a|$ we get that the entire function w converges to 0 as $|z|\rightarrow \infty ,$ a contradiction if $w\not \equiv 0$.

The case $C_{w, \varphi }: Exp^0\rightarrow Exp^0$ is analogous using Lemma 2(ii) and [10, Lemma 1]. $\square$

In the next result we characterize the continuity of the operator for a type of multipliers.

Proposition 6

Consider $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}$ and $w(z)=p_N(z)e^{q_M(z)}$ with $p_N$ and $q_M$ polynomials of degrees N,  $M\in {\mathbb {N}}_0,$ respectively.

• (i) $C_{w,\varphi }: Exp\rightarrow Exp$ is continuous if and only if $M\le 1,$ that is, if $w(z)=p_N(z)e^{\beta z}$ for some $\beta \in {\mathbb {C}}.$

• (ii) $C_{w,\varphi }: Exp^0\rightarrow Exp^0$ is continuous if and only if $w(z)=p_N(z).$

Proof

By Proposition 4, $C_{w,\varphi }$ will be continuous if and only if w belongs to the corresponding space. Given $p_N(z)=\sum _{j=0}^{N}a_jz^j, a_j\in {\mathbb {C}},$ and $q_M(z)=\sum _{j=0}^{M}b_jz^j, b_j\in {\mathbb {C}},$ consider ${\widetilde{p}}_N(z)=\sum _{j=0}^{N}|a_j|z^j$ and ${\widetilde{q}}_M(z)=\sum _{j=0}^{M}|b_j|z^j.$ For $z\in {\mathbb {C}}$ and $\alpha >0$ we have

$$\begin{aligned} |w(z)|e^{-\alpha |z|}=|p_N(z)e^{q_M(z)}|e^{-\alpha |z|}\le {\widetilde{p}}_N(|z|)e^{{\widetilde{q}}_M(|z|)-\alpha |z|}. \end{aligned}$$

So, if $M\le 1$ there exists $n\in {\mathbb {N}}$ such that $w\in H_{v^n}\subseteq Exp.$ If $M=0,$ $w\in H_{v^{1/n}}$ for every $n\in {\mathbb {N}},$ i.e. $w\in Exp^0,$ and the continuity holds. Conversely, if $M\ge 2$ observe that $w\notin H_{v^{\alpha }}$ for each $\alpha >0,$ hence the operator can not be continuous on Exp neither on $Exp^0.$ In the case of $Exp^0,$ if $q_M(z)=b_1 z,$ $b_1\ne 0,$ we can find $c\in {\mathbb {C}},$ $|c|=1,$ and $n\in {\mathbb {N}}$ such that

$$\begin{aligned} \sup _{z\in {\mathbb {C}}}|w(z)|e^{-\frac{1}{n}|z|}=\sup _{z\in {\mathbb {C}}}|p_N(z)e^{b_1z}|e^{-\frac{1}{n}|z|}\ge \sup _{r\ge 0}|p_N(cr)|e^{r(|b_1|-\frac{1}{n})}=\infty . \end{aligned}$$

Thus, as $w\notin H_{v^{1/n}}$ the operator $C_{w, \varphi }$ can not be continuous on $Exp^0.$ $\square$

4 Power Boundedness and Mean Ergodicity

In this section, consider $w\in Exp$ (resp. $w\in Exp^0$) and $\varphi (z)=az+b,$ $a, b \in {\mathbb {C}}.$ We have seen that $C_{w, \varphi }: Exp\rightarrow Exp$ (resp. $C_{w, \varphi }: Exp^0\rightarrow Exp^0$) is continuous. Its iterates have the following expression:

$$\begin{aligned} C_{w,\varphi }^kf=w_{[k]}(f\circ \varphi ^k), \ k\in {\mathbb {N}}, \ f\in Exp \ (Exp^0), \end{aligned}$$

where $w_{[k]}(z):=\prod _{j=0}^{k-1}w(\varphi ^j(z)),$ $z\in {\mathbb {C}}.$ Observe that the symbol $\varphi$ has a fixed point $z_0=\frac{b}{1-a}$ if and only if $a\ne 1,$ and for $k\in {\mathbb {N}}$, we get

$$\begin{aligned} \varphi ^k(z)=\left\{ \begin{array}{ll} a^kz+b\frac{1-a^k}{1-a}=a^k(z-\frac{b}{1-a})+\frac{b}{1-a} &{}\quad \text { if }a\ne 1 \\ z+bk &{}\quad \text { if } a=1. \end{array} \right. \end{aligned}$$

If we take $f\equiv 1,$ 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.

Proposition 7

• (i) If $C_{w, \varphi }: Exp\rightarrow Exp$ is power bounded (resp. mean ergodic), then there exists $m\in {\mathbb {N}}$ such that $\sup _k\Vert w_{[k]}\Vert _{v^m}<\infty$ (resp. $\lim _k\frac{\Vert w_{[k]}\Vert _{v^m}}{k}=0)$.

• (ii) If $C_{w, \varphi }: Exp^0\rightarrow Exp^0$ is power bounded (resp. mean ergodic), then $\sup _k\Vert w_{[k]}\Vert _{v^{1/m}}<\infty$ (resp. $\lim _k\frac{\Vert w_{[k]}\Vert _{v^{1/m}}}{k}=0)$ for every $m\in {\mathbb {N}}.$

In the next result we prove that the necessary condition for power boundedness in Proposition 7 is also sufficient when $C_{\varphi }$ is power bounded on each $H_{v^\alpha },$ $\alpha >0$. This situation occurs for $|a|\le 1$, $a\ne 1$ (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 $C_{\varphi }$ is never mean ergodic on Exp and on $Exp^0$ if $|a|>1$ or if $a=1$ and $b\ne 0.$

Proposition 8

Let $|a|\le 1,$ $a\ne 1$ or $a=1$, $b=0$. The following is satisfied:

• (i) $C_{w, \varphi }: Exp\rightarrow Exp$ is power bounded if and only if there exists $m\in {\mathbb {N}}$ such that $\sup _k\Vert w_{[k]}\Vert _{v^m}<\infty .$

• (ii) $C_{w, \varphi }: Exp^0\rightarrow Exp^0$ is power bounded if and only if $\sup _k\Vert w_{[k]}\Vert _{v^{1/n}}<\infty$ for every $n\in {\mathbb {N}}.$

Proof

Under the hypothesis we are considering, $C_\varphi :H_{v^\alpha }\rightarrow H_{v^\alpha }$ is power bounded for every $\alpha >0$ (see [10, Theorem 22] for the non trivial case $|a|\le 1$, $a\ne 1$, $b\ne 0$).

(i) Assume there exist $m\in {\mathbb {N}}$ and $C>0$ such that $\Vert w_{[k]}\Vert _{v^m}\le C$ for every $k\in {\mathbb {N}}.$ Given $f\in H_{v^n},$ we get

$$\begin{aligned} \Vert C_{w, \varphi }^kf\Vert _{v^{n+m}}=\sup _{z\in {\mathbb {C}}}\prod _{j=0}^{k-1}|w(\varphi ^j(z))|e^{-m|z|}|f(\varphi ^k(z))|e^{-n|z|}\le C\Vert C_{\varphi }^kf\Vert _{v^n} \end{aligned}$$

for every $k\in {\mathbb {N}},$ so the power boundedness holds. The converse follows by Proposition 7. For (ii) proceed analogously in order to get that for every $f\in Exp^0$ and every $n\in {\mathbb {N}}$ there exists $D>0$ such that $\Vert C_{w, \varphi }^kf\Vert _{v^{\frac{1}{n}}}\le D\Vert f\Vert _{v^{\frac{1}{2n}}}$ for every $k\in {\mathbb {N}}.$ $\square$

Since each non constant entire function w is unbounded, it is satisfied that there exists $a\in {\mathbb {C}}$ such that the sequence $\left( \frac{|w(a)|^k}{k}\right) _k$ is unbounded. As an immediate consequence, the next result on multiplication operators is satisfied as follows:

Corollary 9

Let $w\in Exp$ (resp. $Exp^0).$ The multiplication operator $M_{w}: f \mapsto w \cdot f$ is power bounded and (uniformly) mean ergodic on Exp (resp. $Exp^0)$ if and only if $w\equiv \lambda$ with $|\lambda |\le 1$.

In the rest of the paper, for $a\ne 1$ we can assume without loss of generality that $\varphi (z)=az.$ Indeed, in the next lemma we prove that for $a\ne 1,$ every weighted composition operator is conjugated to another one with symbol $\varphi (z)=az.$ So, the study of power boundedness and (uniform) mean ergodicity can be reduced to this case.

Lemma 10

If $a\ne 1,$ $b\in {\mathbb {C}}$ and $X=Exp$ or $X=Exp^0,$ the dynamical systems $C_{w,\varphi }: X\rightarrow X,$ $f\mapsto w(z)f(az+b)$ and $C_{w\circ \tau ,\varphi _a}: X\rightarrow X,$ $f\mapsto w(z-\frac{b}{1-a})f(az),$ where $\tau (z)=z-\frac{b}{1-a}$ and $\varphi _a(z)=az,$ $z\in {\mathbb {C}},$ are conjugated.

Proof

It is easy to see that the conjugacy holds through the homeomorphism $C_{\tau }: X\rightarrow X,$ $f(z)\mapsto f(z-\frac{b}{1-a}).$ Indeed, $C_{\tau }^{-1}\circ C_{w\circ \tau , \varphi _a}\circ C_{\tau } =C_{w, \varphi }.$ $\square$

Proposition 11

Consider $\varphi (z)=az,$ $a\in {\mathbb {C}},$ if $a\ne 1$ and $\varphi (z)=z+b,$ $b\in {\mathbb {C}}{\setminus }\{0\}$ otherwise. The operator $C_{w, \varphi }$ is not mean ergodic and thus not power bounded on Exp and on $Exp^0$ if any of the following conditions is satisfied:

• (i) $a\ne 1$ and $|w(0)|> 1.$

• (ii) $a=1$ and there exist $n_0\in {\mathbb {N}}$ and $C>1$ such that $\left| w(jb)\right| >C$ for every $j\in {\mathbb {N}}, j\ge n_0.$

• (iii) $|a|>1$ and there exist $n_0\in {\mathbb {N}}$ and $C>0$ satisfying $|w(a^j)|>C$ for every $j\in {\mathbb {N}}, j\ge n_0.$

• (iv) $a=1$ or $|a|>1$ and there exists an increasing subsequence $(k_s)_s\subseteq {\mathbb {N}}$ satisfying $\prod _{j=0}^{k_s-1}|w(\varphi ^j(z_0))|>C$ for some $z_0\in {\mathbb {C}},$ $C>0,$ and every $s\in {\mathbb {N}}.$

Proof

• (i) and (ii) follow by Proposition 7, as $\Vert w_{[k]}\Vert _{v^{\alpha }}\ge \prod _{j=0}^{k-1}\left| w\left( \varphi ^j(z_0) \right) \right| e^{-\alpha |z_0|}$ tends to infinity for $z_0=0$ and $z_0=n_0b,$ respectively.

• (iii) Take $M\in {\mathbb {N}}$ such that $|a|^{M}>1/C$ and consider $f(z)=z^M.$ Evaluating at the point $z_0=a^{n_0},$ for every $\alpha >0$ we get

  $$\begin{aligned} \frac{\Vert C^k_{w, \varphi }f\Vert _{v^{\alpha }}}{k}\ge \frac{1}{k}\prod _{j=0}^{k-1}|w(a^{j+n_0})||f(a^{k+n_0})|e^{-\alpha |z_0|}> |a|^{n_0M}e^{-\alpha |z_0|}\frac{(C|a|^M)^k}{k}. \end{aligned}$$

  So, the operator can be mean ergodic neither on Exp,  nor on $Exp^0$. (iv) For $f(z)=z$ and $\alpha >0$ we get

  $$\begin{aligned} \frac{\Vert C^{k_s}_{w, \varphi }f\Vert _{v^{\alpha }}}{k_s}\ge \frac{1}{k_s}\prod _{j=0}^{k_s-1}|w(\varphi ^j(z_0))||\varphi ^{k_s}(z_0)|e^{-\alpha |z_0|}> Ce^{-\alpha |z_0|}\frac{|\varphi ^{k_s}(z_0)|}{k_s}. \end{aligned}$$

  As $\varphi ^{k_s}(z_0)= z_0a^{k_s}$ if $|a|>1$ and $\varphi ^{k_s}(z_0)=z_0+k_s b$ if $a=1,$ the conclusion holds.

$\square$

We have seen above that a necessary condition for the power boundedness and mean ergodicity of $C_{w, \varphi }$ 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 $w(z)=p_N(z)e^{\beta z},$ $N\in {\mathbb {N}}_0,$ $\beta \in {\mathbb {C}}$ when considering the space Exp and $w(z)=p_N(z),$ $N\in {\mathbb {N}}_0,$ when considering $Exp^0,$ where $p_N$ is a polynomial of degree N. For $\alpha >0$ we get the following:

If $a\ne 1,$

$$\begin{aligned} \Vert w_{[k]}\Vert _{v^{\alpha }}=\sup _{z\in {\mathbb {C}}}\left( \prod _{j=0}^{k-1}|p_N(\varphi ^j(z))|\right) |e^{\beta \frac{1-a^k}{1-a}z}|e^{-\alpha |z|} \end{aligned}$$

(4.1)

If $a=1,$ $b\ne 0,$

$$\begin{aligned} \Vert w_{[k]}\Vert _{v^{\alpha }}= & {} |e^{\beta b (k-1)/2}|^k\sup _{z\in {\mathbb {C}}}\left( \prod _{j=0}^{k-1}|p_N(z+jb)|\right) |e^{\beta k z}|e^{-\alpha |z|} \end{aligned}$$

(4.2)

4.1 Case $|a|<1$

Lemma 12

Let $s,t>0$ and $f_s(x):=x^se^{-tx}$, $x\in [0,\infty )$. The maximum of $f_s$ is attained at $x_s=s/t$ and $f_s(x_s)=(\frac{s}{et})^s$ satisfies that $\lim _{s\rightarrow \infty } x_s =\infty$, $\lim _{s\rightarrow \infty } M^sf_s(x_s)=\infty$ and $\lim _{s\rightarrow \infty } a^{s^2} M^s f_s(x_s)=0$ for each $0<a<1$ and $M>0$.

Theorem 13

Let $\varphi (z)=az, |a|<1.$ If $w(z)=p_N(z)e^{\beta z},$ $N\in {\mathbb {N}}_0,$ $\beta \in {\mathbb {C}}$ when considering $C_{w, \varphi }: Exp\rightarrow Exp$ and $\beta =0$ for $C_{w, \varphi }:Exp^0\rightarrow Exp^0,$ then $C_{w, \varphi }$ is power bounded and (uniformly) mean ergodic if and only if $|w(0)|\le 1.$

Proof

If $|w(0)|>1,$ the operator cannot be mean ergodic by Proposition 11(i).

If $|w(0)|\le 1,$ [13, Theorem 3.10(i)] yields that for $r>0,$ there exists $C>1$ such that, for every $|z|\le r$ and $k\in {\mathbb {N}},$ then $\prod _{j=0}^{k}|w(a^jz)|<C.$ On the other hand, observe that there exists $M>0$ such that $|p_N(z)|\le M |z|^N$ for every $|z|\ge r.$ Fix $z_0\in {\mathbb {C}}$ and take $j_0\in {\mathbb {N}}$ such that $|a^jz_0|\le r$ for every $j\in {\mathbb {N}}, j\ge j_0.$ Therefore, for $k\ge j_0$ we have

$$\begin{aligned} \prod _{j=0}^{k-1}|p_N(a^jz_0)e^{\beta a^jz_0}|\le & {} \prod _{j=0}^{j_0-1}M\left| a^jz_0|^N e^{|\beta | |z_0| \frac{1-|a|^{j_0}}{1-|a|}}\right| \prod _{j=j_0}^{k}|w(a^jz_0)|\\\le & {} CM^{j_0}|z_0|^{Nj_0}|a|^{N j_0(j_0-1)/2}e^{\frac{|\beta | |z_0|}{1-|a|}} . \end{aligned}$$

Consider $\alpha > \frac{|\beta |}{1-|a|}$ and $k\in {\mathbb {N}}.$ For $s= \min (j_0, k),$ we get the following:

$$\begin{aligned} \prod _{j=0}^{k-1}|w(a^jz_0)|e^{-\alpha |z_0|}\le C M^{s}|a|^{N s(s-1)/2}\sup _{z\in {\mathbb {C}}}|z|^{Ns}e^{-(\alpha - \frac{|\beta |}{1-|a|}) |z|} \end{aligned}$$

By Lemma 12, there exists $D>0,$ independent of $z_0,$ such that

$$\begin{aligned} \prod _{j=0}^{k-1}|w(a^jz_0)|e^{-\alpha |z_0|}\le D \end{aligned}$$

for every $k\in {\mathbb {N}}.$ So, the conclusion holds by Proposition 8. $\square$

4.2 Case $|a|=1,$ $a\ne 1$

Theorem 14

Given $\varphi (z)=az, |a|=1,$ $a\ne 1,$ we get the following:

• (i) If $w(z)=p_N(z)e^{\beta z},$ $N\ne 0,$ $\beta \in {\mathbb {C}}$ when considering $C_{w, \varphi }:Exp\rightarrow Exp$ and $\beta =0$ for $C_{w, \varphi }:Exp^0\rightarrow Exp^0,$ the operator is not mean ergodic, hence not power bounded.

• (ii) If $w(z)=\lambda e^{\beta z},$ $\lambda , \beta \in {\mathbb {C}}$ when considering $C_{w, \varphi }:Exp\rightarrow Exp$ and $\beta =0$ for $C_{w, \varphi }:Exp^0\rightarrow Exp^0,$ the operator is power bounded, thus uniformly mean ergodic if and only if $|\lambda |\le 1$.

Proof

• (i) Observe that there exist $R>0$ and $M>0$ such that $|p_N(z)|>M|z|^N$ for every $|z|>R.$ If $N\ne 0$ and $k\in {\mathbb {N}}$, then

  $$\begin{aligned}&\Vert w_{[k]}\Vert _{v^{\alpha }}>M^k\sup _{|z|>R}|z|^{Nk}|e^{\beta \frac{1-a^k}{1-a}z}|e^{-\alpha |z|}\ge M^k\sup _{r> R}r^{Nk}e^{-(\alpha -|\beta | \frac{|1-a^k|}{|1-a|})r}\\&\quad \ge M^k\sup _{r> R}r^{Nk}e^{-\alpha r}. \end{aligned}$$

  By Lemma 12, if k is big enough, then

  $$\begin{aligned} \sup _{r> R}r^{Nk}e^{-\alpha r}=\sup _{r> 0}r^{Nk} e^{-\alpha r}. \end{aligned}$$

  Thus, the operator cannot be mean ergodic and power bounded by applying again Lemma 12 and Proposition 8.

• (ii) The case $|\lambda |>1$ follows by Proposition 11(i), since $|w(0)|>1.$ If $|\lambda |\le 1,$ for every $\alpha \ge \frac{2|\beta |}{|1-a|}$ we get

  $$\begin{aligned} \Vert w_{[k]}\Vert _{v^{\alpha }}\le \sup _{z\in {\mathbb {C}}}e^{-(\alpha -\frac{2|\beta |}{|1-a|})|z|}=1. \end{aligned}$$

  Now, the conclusion follows by Proposition 8.

$\square$

4.3 Case $a=1$

Theorem 15

Let $\varphi (z)=z+b, \ b\in {\mathbb {C}}{\setminus }\{0\}.$

• (i) If $w(z)=p_N(z)e^{\beta z},$ $\beta \in {\mathbb {C}},$ $N\in {\mathbb {N}}_0,$ then the operator $C_{w, \varphi }$ is not mean ergodic and thus not power bounded on Exp.

• (ii) If $w(z)=p_N(z),$ $N\in {\mathbb {N}}_0,$ the operator $C_{w, \varphi }$ is power bounded and mean ergodic on $Exp^0$ if and only if $w\equiv \lambda ,$ $|\lambda |<1.$

Proof

If $\beta \ne 0,$ given $\alpha >0$ we can find $k\in {\mathbb {N}}$ such that $|\beta | k>\alpha$ and $\theta _k\in {\mathbb {R}}$ such that the supremum in (4.2) is bigger than

$$\begin{aligned} \sup _{r\ge 0}\prod _{j=0}^{k-1}|p_N(re^{i\theta _k}+jb)|e^{r(|\beta | k-\alpha )}=\infty . \end{aligned}$$

Therefore, $w_{[k]}\notin H_{v^{\alpha }}$ and the operator is not mean ergodic and not power bounded on Exp by Proposition 7. If $\beta =0$ and $N\ne 0,$ there exist $n_0$ and $C>1$ such that $\left| p_N(jb)\right| >C$ for $j>n_0.$ Now, the conclusion holds by Proposition 11(ii).

Finally, let us study the case $w\equiv \lambda ,$ $\lambda \in {\mathbb {C}}.$ For $\lambda C_{\varphi }: Exp\rightarrow Exp,$ take $c\in {\mathbb {C}}$ such that $cb=|c||b|$ and $|\lambda |>e^{-|cb|}.$ The function $e^{cz}\in Exp$ and it satisfies $(\lambda C_{\varphi })^ke^{cz}=(\lambda e^{cb})^ke^{cz}.$ So, as $|\lambda e^{cb}|>1,$ we cannot find $n\in {\mathbb {N}}$ such that $\frac{(\lambda C_{\varphi })^k(e^{cz})}{k}$ is bounded on $H_{v^n};$ therefore, the operator is not mean ergodic on Exp.

In the case of $\lambda C_{\varphi }: Exp^0\rightarrow Exp^0,$ the operator is neither power bounded, nor mean ergodic if $|\lambda |>1$ by Proposition 7(ii). If $|\lambda |=1,$ take $f(z)=z\in Exp^0$ in order to see that, evaluating at $z=0,$

$$\begin{aligned} \frac{\Vert (\lambda C_{\varphi })^kf\Vert _{v^{1/n}}}{k}=\frac{1}{k}\sup _{z\in {\mathbb {C}}}|z+kb|e^{-\frac{|z|}{n}}\ge |b|. \end{aligned}$$

Therefore, the operator cannot be mean ergodic, neither power bounded. If $|\lambda |<1,$ there exists $n_0\in {\mathbb {N}}$ such that $|\lambda |< e^{-\frac{|b|}{n}}$ for every $n\ge n_0.$ [10, Theorem 19(i)] implies the operator $\lambda C_{\varphi }:H_{v^{1/n}}\rightarrow H_{v^{1/n}}$ is power bounded for every $n\ge n_0,$ thus it is power bounded and uniformly mean ergodic on $Exp^0.$ $\square$

4.4 Case $|a|>1$

Theorem 16

Given $\varphi (z)=az, |a|>1,$ and $w(z)=p_N(z)e^{\beta z},$ $\beta \in {\mathbb {C}},$ $N\in {\mathbb {N}}_0,$ the operator $C_{w, \varphi }$ is not mean ergodic and thus not power bounded on Exp and on $Exp^0$ (when considering $Exp^0$, put $\beta =0)$.

Proof

Take $n_0$ such that $|p_N(a^k)|>1$ for every $k\ge n_0,$ $z_0=a^{n_0}$ and consider $(k_s)_s\subseteq {\mathbb {N}}$ increasing such that Arg($\beta a^{n_0}\frac{1-a^{k_s}}{1-a})=$Arg($\frac{\beta a^{n_0}}{1-a}$)+Arg$(1-a^{k_s})\in [-\frac{\pi }{2}, \frac{\pi }{2}].$ Then, for every $s\in {\mathbb {N}},$

$$\begin{aligned}&\prod _{j=0}^{k_s-1}|w(a^{j+n_0})|=\prod _{j=0}^{k_s-1}|p_N(a^{j+n_0})||e^{\beta a^{j+n_0}}|>|e^{\beta a^{n_0}\frac{1-a^{k_s}}{1-a}}|\\&=e^{\text {Re}(\beta a^{n_0}\frac{1-a^{k_s}}{1-a})}\ge 1. \end{aligned}$$

Then, Proposition 11(iv) yields the following conclusion: $\square$

Remark 17

Theorems 15(i) and 16 when $w\equiv \lambda$, $|\lambda |<1,$ show that Proposition 8 does not hold in general.

5 Multiples of Composition Operators

In this section we focus on multiples of composition operators, that is, operators of the form $\lambda C_{\varphi },$ $\lambda \in {\mathbb {C}},$ $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}.$ 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 $Exp^0.$

5.1 Spectrum

Regarding the invertibility of the weighted composition operator $C_{w, \varphi },$ $\varphi (z)=az+b,$ on Exp and $Exp^0,$ we observe important differences from the results obtained on the Banach spaces $H_{v^{\alpha }},$ $\alpha >0$. Notice that $C_{w, \varphi }^{-1}=C_{\frac{1}{w\circ \varphi ^{-1}}, \varphi ^{-1}},$ with $\varphi ^{-1}(z)=\frac{1}{a}z-\frac{b}{a},$ $z\in {\mathbb {C}}.$ As $C_{w, \varphi }: H_{v^{\alpha }} \rightarrow H_{v^{\alpha }}$ can only be continuous if $|a|\le 1$ (see [10, Theorem 8]), it has no inverse, i.e. $0\in \sigma (C_{w, \varphi }, H_{v^{\alpha }})$ if $|a|<1.$ By contrast, Proposition 4 yields the operator can be invertible when acting on Exp and $Exp^0:$

Proposition 18

Let $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}.$ The operator $C_{w, \varphi }: Exp\rightarrow Exp$ (resp. $C_{w, \varphi }: Exp^0\rightarrow Exp^0$) is invertible, i.e. $0\notin \sigma (C_{w, \varphi })$ if and only if w and $\frac{1}{w}$ belong to Exp (resp. $Exp^0$). In particular, $0\notin \sigma (\lambda C_{\varphi }), \lambda \in {\mathbb {C}}.$

Lemma 19

The entire function $f(z)=\sum _{j=0}^{\infty }a_jz^j, a_j, z\in {\mathbb {C}},$ satisfies the following:

• (i) It belongs to Exp if and only if there exists $B, C>0$ such that $|a_j|\le C\frac{B^j}{j!}$ for every $j\in {\mathbb {N}}.$

• (ii) It belongs to $Exp^0$ if and only if, for every $B>0$ there exists $C>0$ such that $|a_j|\le C\frac{1}{B^jj!}$ for every $j\in {\mathbb {N}}.$

Proof

(i) is proved in [21, Lemma 4.18] and (ii) follows analogously. $\square$

The study of the spectrum of multiples of composition operators reduces to the study of the spectrum of $C_{\varphi }.$ In the next proposition we determine it. Compare the result to [10, Proposition 13] and [20, Corollary 8 and Theorem 5].

Proposition 20

For the composition operator $C_{\varphi },$ $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}},$ we get the follows:

• (i) If $a=1,$ then $\sigma (C_{\varphi }, Exp)=\sigma _p( C_{\varphi }, Exp)={\mathbb {C}}{\setminus } \{0\}$ and $\sigma ( C_{\varphi }, Exp^0)=$ $\sigma _p (C_{\varphi }, Exp^0)=\{1\}.$

• (ii) If $|a|\ne 1$ or $a\ne 1$ is a root of unity, then $\sigma _p (C_{\varphi })=\sigma (C_{\varphi })=\{a^j, j=0, 1, \dots \}$.

• (iii) if $|a|=1$ and a is not a root of unity, then $\sigma _p (C_{\varphi })=\{a^j, j=0, 1, \dots \}$ and $\mu \in \sigma (C_{\varphi })$ if and only if for all $k>0$ and $\varepsilon >0,$ there exists $n\in {\mathbb {N}}$ such that $|a^n-\mu |< \varepsilon k^{-n}.$

Proof

• (i) Consider $a=1$ and $C_{\varphi }: Exp \rightarrow Exp.$ It is easy to see that the function $e^{\alpha z}$ is an eigenvector associated with the eigenvalue $e^{\alpha b}$ for every $\alpha \in {\mathbb {C}}.$ Thus, ${\mathbb {C}}{\setminus } \{0\}\subseteq \sigma _p(C_{\varphi }, Exp)\subseteq \sigma (C_{\varphi }, Exp)$ and the conclusion is satisfied by Proposition 18. When considering $C_{\varphi }: Exp^0 \rightarrow Exp^0,$ as $Exp^0={{\,\mathrm{\mathrm{proj}}\,}}_{n\ge m}H_{v^{1/n}}$ for every $m\in {\mathbb {N}},$ [3, Lemma 2.1] and [10, Proposition 13] yield $\sigma (C_{\varphi }, Exp^0)\subseteq \cup _{n\ge m} \sigma (C_{\varphi }, H_{v^{1/n}})=\{ e^{\delta }, |\delta |\le \frac{|b|}{m}\}$ for every $m\in {\mathbb {N}}$. Thus, $\sigma (C_{\varphi }, Exp^0)\subseteq \{1\}.$ 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 $a\ne 1,$ then $C_{\varphi }z^j=a^jz^j,$ $j\in {\mathbb {N}}_0,$ and the proof of [22, Proposition 3.3(i)]. Let us analyse the spectrum. First, observe that if $|a|\le 1,$ $a\ne 1,$ then $\{a^j, j=0, 1, \dots \}\subseteq \sigma (C_{\varphi })\subseteq \overline{\{a^j, j=0, 1, \dots \}}$ by [3, Lemma 2.1], [4, Lemma 5.2] and [10, Proposition 13].

• (ii) By the inclusions above, if $a\ne 1$ is a root of unity the conclusion is trivial, and if $|a|<1,$ the assertion holds because $0\notin \sigma (C_{\varphi })$ by Proposition 18. If $|a|>1,$ as $\sigma (C_{\varphi })=(\sigma (C_{\varphi ^{-1}}))^{-1}$ with $\varphi ^{-1}(z)=\frac{1}{a}z,$ the previous case yields the following conclusion.

• (iii) Assume $|a|=1$ and a is not a root of unity. We prove that $\mu \in \rho (C_{\varphi })$ if and only if there exists $k>0$ and $\varepsilon >0$ such that $|a^j-\mu |\ge \varepsilon k^{-j}$ for each $j\in {\mathbb {N}}.$

  Let us analyse first $\sigma (C_{\varphi }, Exp).$ We have seen above that, if the condition is satisfied, $C_{\varphi }-\mu I$ is injective. We prove it is also surjective. Given $g(z)=\sum _{j=0}^{\infty }a_jz^j\in Exp,$ by Lemma 19(i) there exists $B, C>0$ such that $|a_j|\le C\frac{B^j}{j!}$ for every $j\in {\mathbb {N}}.$ It is easy to see that the function $f(z):=\sum _{j=0}^{\infty }\frac{a_j}{a^j-\mu }z^j, z\in {\mathbb {C}},$ satisfies $(C_{\varphi }-\mu I)f=g$ and that there exists $D>0$ such that

  $$\begin{aligned} \frac{|a_j|}{|a^j-\mu |}\le C\frac{B^j k^j}{j!\varepsilon }\le D\frac{(Bk)^j}{j!}. \end{aligned}$$

  Again by Lemma 19, we get $f\in Exp,$ and so, $\mu \in \rho (C_{\varphi }, Exp).$ For the converse, assume $\mu \in \rho (C_{\varphi }, Exp).$ Then $(C_{\varphi }-\mu I)^{-1}: Exp\rightarrow Exp$ exists, it is continuous and satisfies

  $$\begin{aligned} (C_{\varphi }-\mu I)^{-1}\left( \sum _{j=0}^{\infty }a_jz^j\right) =\sum _{j=0}^{\infty }\frac{a_j}{a^j-\mu }z^j. \end{aligned}$$

  The continuity implies (see Lemma 1) that for $n=1$ there exist $k\in {\mathbb {N}}$ and $C>0$ such that, for each monomial $z^j\in Exp,$ $j\in {\mathbb {N}}_0,$ we have

  $$\begin{aligned} \sup _{z\in {\mathbb {C}}} \frac{|z|^j}{|a^j-\mu |}e^{-k|z|}\le C \sup _{z\in {\mathbb {C}}} |z|^je^{-|z|}. \end{aligned}$$

  By Lemma 12, for every $j\in {\mathbb {N}}_0$ we get

  $$\begin{aligned} \left( \frac{j}{k e}\right) ^j\frac{1}{|a^j-\mu |}\le C \left( \frac{j}{e}\right) ^j, \text { i.e. } |a^j-\mu |\ge \frac{\varepsilon }{k^j} \text { for some } \varepsilon >0. \end{aligned}$$

  For $\sigma (C_{\varphi }, Exp^0)$ proceed analogously but using Lemma 19(ii) and Lemma 2.

$\square$

5.2 Dynamics

As an immediate consequence of the theorems in Sect. 4, we get the following:

Corollary 21

Let $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}}.$ The properties of (uniform) mean ergodicity and power boundedness are equivalent for the operator $\lambda C_{\varphi },$ $\lambda \in {\mathbb {C}},$ acting on Exp and on $Exp^0.$ The following is satisfied:

• (i) $\lambda C_{\varphi }: Exp\rightarrow Exp$ is mean ergodic (power bounded) if and only if $|a|\le 1, a\ne 1$ and $|\lambda |\le 1.$

• (ii) $\lambda C_{\varphi }: Exp^0\rightarrow Exp^0$ is mean ergodic (power bounded) if and only if $|a|\le 1, a\ne 1$ and $|\lambda |\le 1,$ or $a=1$ and $|\lambda |< 1.$

The last characterization differs from the one obtained for the Banach space $H_{v^{\alpha }}^0,$ $\alpha >0,$ if $|a|= 1$ and a is not a root of unity or if $a=1$ (recall that on $H_{v^{\alpha }}^0$ the continuity of the operator only occurs when $|a|\le 1$). In this setting we get that for $|a|=1,$ a not a root of unity and $|\lambda |=1,$ the multiples of composition operators are power bounded and mean ergodic on $H_{v^{\alpha }}^0,$ but not uniformly mean ergodic. When $a=1$ and $\lambda$ is small enough, the operator can be power bounded and uniformly mean ergodic, unlike in Exp. However, we can find $\lambda ,$ $|\lambda |<1,$ such that the operator is neither power bounded nor mean ergodic on the Banach setting, unlike what happens in $Exp^0$ (see [10]).

We finish this section studying the hypercyclicity of the operator. It is considered only in the case $a=1,$ since otherwise $\varphi (z)=az+b$ has a fixed point and [15, Proposition 2.1] yields $C_{w,\varphi }$ cannot be weakly supercyclic. So, we focus on multiples of translation operators, that is, on operators of the form $\lambda T_b,$ $\lambda , b\in {\mathbb {C}},$ $\lambda T_b(f)(z)=\lambda f(z+b).$ The result is a consequence of the one obtained for $H_{v^{\alpha }},$ $\alpha >0$ in [10, Theorem 20].

Theorem 22

The weighted translation operator $\lambda T_b,$ $\lambda , b \in {\mathbb {C}},$ satisfies the following:

• (i) It is hypercyclic, topologically mixing and chaotic on Exp for every $\lambda \in {\mathbb {C}}.$

• (ii) It is hypercyclic on $Exp^0$ if and only if $|\lambda |=1.$ In this case, it is topologically mixing.

Proof

(i) Given $\lambda \in {\mathbb {C}}$ there exists $m\in {\mathbb {N}}$ such that $e^{-m|b|}<|\lambda |<e^{m|b|}.$ By [10, Theorem 20], $\lambda T_b: H_{v_{m}}^0\rightarrow H_{v_{m}}^0$ is topologically mixing (thus, hypercyclic) and chaotic. By the comparison principle (see [17, Lemma 3]), the operator is hypercyclic, topologically mixing and chaotic on Exp. This assertion also follows by [11, Proposition 3.3].

(ii) Given $\lambda \in {\mathbb {C}},$ $|\lambda |\ne 1,$ there exists $m\in {\mathbb {N}}$ such that $|\lambda |< e^{-\frac{|b|}{m}}$ or $e^{\frac{|b|}{m}}<|\lambda |.$ By [10, Theorem 20], $\lambda T_b: H_{v_{1/m}}^0\rightarrow H_{v_{1/m}}^0$ is not hypercyclic. As $Exp^0$ is dense on $H_{v_{1/m}}^0,$ the comparison principle yields that $\lambda T_b$ cannot be hypercyclic on $Exp^0.$ Consider now $|\lambda |=1.$ Without loss of generality, we can consider $\lambda =1$ [9, Corollary 3.3]. Now the conclusion follows by [11, Proposition 3.7]. $\square$

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)].

Theorem 23

Let $\varphi (z)=az+b$ $a, b\in {\mathbb {C}},$ $|a|<1$ and $w(z)=p_N(z)e^{\beta z},$ $N\in {\mathbb {N}}_0,$ $\beta \in {\mathbb {C}}.$ If $|w(0)|\le 1,$ then $C_{w, \varphi }: H_{v^{\alpha }}\rightarrow H_{v^{\alpha }}$ and $C_{w, \varphi }: H_{v^{\alpha }}^0\rightarrow H_{v^{\alpha }}^0$ are power bounded and uniformly mean ergodic for every $\alpha >\frac{|\beta |}{1-|a|}.$

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 $C_{w, \varphi }$ is power bounded and compact on $H_{v^{\alpha }}$ and on $H_{v^{\alpha }}^0$ for every $\alpha >\frac{|\beta |}{1-|a|}.$ As a consequence, $C_{w, \varphi }$ is uniformly mean ergodic on these spaces by Yosida-Kakutani Mean Ergodic Theorem [30]. $\square$

The next result completes [10, Theorem 16(b)]:

Theorem 24

Let $\varphi (z)=az+b,$ $a, b\in {\mathbb {C}},$ $|a|<1$ and $w(z)=e^{\beta z},$ $\beta \in {\mathbb {C}}.$ The operator $C_{w, \varphi }: H_{v^{\alpha }}^0\rightarrow H_{v^{\alpha }}^0,$ $\alpha =\frac{|\beta |}{1-|a|},$ is power bounded. It is mean ergodic if and only if $a\in {\mathbb {D}}{\setminus } [0, 1).$

Proof

By [10, Theorem 16(b)], it only remains to show that for $a\in {\mathbb {D}}{\setminus } [0, 1),$ the operator is mean ergodic. Without loss of generality, assume $b=0.$ Observe that for every $f\in H_{v^{\alpha }}^0$ and every $z\in {\mathbb {C}},$ $C_{w, \varphi }^kf(z)=e^{\beta z \frac{1-a^k}{1-a}}f(a^kz)\overset{k\rightarrow \infty }{\longrightarrow } e^{\frac{\beta z}{1-a}}f(0)\in H_{v^{\alpha }}^0$ (observe that $e^{\frac{\beta z}{1-a}}f(0)\notin H_{v^{\alpha }}^0$ if $a\in (0, 1)).$ Proceeding as in the proof of [23, Lemma 3.4(1)], we get $C_{w, \varphi }^kf$ converges to $e^{\frac{\beta z}{1-a}}\delta _0(f)$ 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].$\square$