This paper focuses on mean ergodicity of composition operators between given weighted Banach spaces of analytic functions defined on the unit disk and Hardy spaces of unit disk.

For a locally convex Hausdorff space, the space of all continuous linear operators from X into itself is denoted by $L(X)$. Equipping $L(X)$ with its strong operator topology, we write $L_{s}(X)$. Given $T\in L(X)$, its Cesàro means (see [1]) are defined by

$ \begin{eqnarray}\label{def ceraro means} T_{[n]}:=\frac{1}{n}\sum\limits_{m=1}^{n}T^{m}, \ \ n\in \mathbb{N}\end{eqnarray} $

(1.1)

from which one could routinely verify

$ \begin{eqnarray}\label{def Tn} \frac{1}{n}T^{n}=T_{[n]}-\frac{(n-1)}{n}T_{[n-1]}, \ \ n\in \mathbb{N}, \end{eqnarray} $

(1.2)

where $T_{[0]}$ is the identity operator on $X$.

An operator $T$ is mean ergodic if $\{T_{[n]}\}_{n=0}^{\infty}$ is a convergent sequence in $L_{s}(X)$ (see [1]).

Motivated by [1] in which the mean ergodicity of multiplication operators in weighted Banach spaces of holomorphic functions is examined, in this paper, the mean ergodicity of composition operators between given weighted Banach spaces of analytic functions defined on the unit disk and Hardy spaces of unit disk will be examined.

The classic space $H^{\infty}$ is the space of all bounded analytic functions $f$ on on the unit disc $\mathbb{U}=\{z\in \mathbb{C}:|z|< 1\}$ endowed with the norm

$ \begin{eqnarray}\label{def H infinity } \|f\|_{\infty}=\sup\limits_{|z|<1}|f(z)|.\end{eqnarray} $

(2.3)

Let $S$ denote the subset of $H^{\infty}$ consisting of the analytic selfmaps of $\mathbb{U}$.

For every $C_{w}$ denotes the composition operators of constant symbol $w$. For every $n=1, 2, \cdots$, denote $\varphi^{[n]}=\varphi\circ\cdots\circ\varphi$, then $C_{\varphi^{[n]}}=\varphi^{[n]}$.

Let $v$ be strictly positive bounded continuous functions (weights) on $\mathbb{U}$. We are interested in radial weights, i.e., weights which satisfy $v(z)=v(|z|)$. Especially interesting ones are weights $v$ which satisfy the following condition

$ \begin{eqnarray}\label{L1 weig} \exists q>0: \frac{1}{v(1-\frac{1}{t})t^{q}} {\rm\ \ is \ almost\ increasing}, \ t\geq1. \end{eqnarray} $

(2.4)

Let $H_{v}^{\infty}$ denote the Banach spaces of analytic functions defined on $\mathbb{U}$ endowed with the norm

$ \begin{eqnarray}\label{def Hv infinity } \|f\|_{v}=\sup\limits_{|z|<1}v(z)|f(z)|.\end{eqnarray} $

(2.5)

The boundedness and weak compactness of composition operator $C_{\varphi}f=f\circ \varphi $ on $H_{v}^{\infty}$ were investigated by several authors. We refer to [2] and [10].

In this section the mean ergodicity of composition operators between the spaces $H_{v}^{\infty}$ is considered. To present the result, first we need some auxiliary results. Recall that for any $z \in \mathbb{U}$, $\varphi_{z}(w)$ is the Màbius transformation of $ \mathbb{U}$ which interchanges the origin and $z$, i.e.,

$ \begin{eqnarray*}\varphi_{z}(w)=\frac{z-w}{1-\overline{z}w}, \ w\in \mathbb{U}.\end{eqnarray*} $

The psudohyperbolic distance $\rho (z, w)$ for all $z, w \in \mathbb{U}$ is defined by $\rho (z, w)=\big|\frac{z-w}{1-\overline{z}w}\big|$.

For $z, w \in \mathbb{U}$, the hyperbolic distance from $z$ to $w$ to be the "length of the shortest curve from $z$ to $w$", that is

$ \rho_{\mathbb{U}}(z, w):=\inf\limits_{\gamma}\ell_{\mathbb{U}}(\gamma), $

where on the right, $\gamma$ runs through all piecewise $C^{1}$ curve from $z$ to $w$ (see p.151 in [3] for details).

The relation between the psudohyperbolic distance and hyperbolic distance is reflected by the following:

Lemma 2.1   For $z, w \in \mathbb{U}$ we have

$ \rho_{\mathbb{U}}(z, w)=\ell_{\mathbb{U}}(\gamma) =\log\frac{1+\rho (z, w)}{1-\rho (z, w)}, $

where $\gamma$ is the unique arc that joins $z$ and $w$, and lies on a circle perpendicular to the unit circle.

Proof  See p.153 in [3].

Lemma 2.2  Let $\varphi$ be a holomorphic self-map of $\mathbb{U}$ with a fixed point $p$, then for any $z\in \mathbb{U}$,

$ \begin{eqnarray}\label{6} \rho _{\mathbb{U}}(\varphi^{[n]}(z), p)\leq \|\varphi\|^{n}_{\infty}\rho _{\mathbb{U}}(z, p) \end{eqnarray} $

(2.6)

for all $z, p \in \mathbb{U}$.

Proof  From Exercises 5-7 on p.171 in [3], we have

$ \begin{eqnarray*} \rho _{\mathbb{U}}(\varphi(z), \varphi(p))\leq \|\varphi\|_{\infty}\rho _{\mathbb{U}}(z, p) \end{eqnarray*} $

for all $z \in \mathbb{U}$. Since $p$ is a fixed point of $\varphi$, we have $\varphi^{[n]}(p)=p$ for all $n=1, 2, 3, \cdots$. Iteration yields

$ \begin{eqnarray*} \rho (\varphi^{[n]}_{\mathbb{U}}(z), p)\leq \|\varphi\|^{n}_{\infty}\rho_{\mathbb{U}} (z, p). \end{eqnarray*} $

Lemma 2.3  Let $v$ be a weight such that $v$ is radial and satisfying (2.4). For every $f \in H_{v}^{\infty}$ there exists a constant $C$ (depending on the weight $v$) such that

$ \begin{eqnarray}\label{est f} |f(z)-f(p)|\leq C_{v}\|f\|_{v}\max\big\{\frac{1}{v(z)}, \frac{1}{v(p)}\big\}\rho (z, p) \end{eqnarray} $

(2.7)

for all $z, p \in \mathbb{U}$.

Proof  Adaptation of the case $N=1$ in Lemma 3.2 from [10] gives the proof.

The main result of this section is as follows:

Theorem 2.1  Let $v$ be a weight such that $v$ is radial and satisfies (2.4). If $\varphi\in $ S is a holomorphic self-map of $\mathbb{U}$ with a fixed point, satisfying

$ \begin{eqnarray}\label{HV ergodic standard}\lim\limits_{n\rightarrow\infty}\|\varphi\|^{n}_{\infty}=0, \end{eqnarray} $

(2.8)

then $C_{\varphi}$ is mean ergodic in $H^{v}_{\infty}$.

Proof  For any $f(z)$ holomorphic in $\mathbb{U}$, denote $C_{p}f=f(p)$ where $p$ is the fixed point of $\varphi$. It is obviously that $C_{p}$ is a bounded composition operator. We are going to verify

$ \begin{eqnarray}\label{verify ergodic }\|C_{[n]}-C_{p}\|_{v}\rightarrow 0. \end{eqnarray} $

(2.9)

For any $f\in H_{v}^{\infty}$, $\|f\|_{v}=1$, by (2.7) in Lemma 2.3, we have

$ \|C_{\varphi^{[n]}}f(z)-C_{p}f(z)\|_{v}=\sup\limits_{|z|<1}v(z)|f(\varphi^{[n]}(z))-f(p)| \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq \sup\limits_{|z|<1}v(z)C_{v}\max\big\{\frac{1}{v(\varphi^{[n]}(z))}, \frac{1}{v(p)}\big\}\rho (\varphi^{[n]}(z), p). $

Combining Lemma 2.1 with (2.6) in Lemma 2.2 yields

$ \begin{eqnarray}\label{single est Hv norm} \|C_{\varphi^{[n]}}f(z)-C_{p}f(z)\|_{v}\leq C'_{v} \|\varphi\|^{n}_{\infty}, \end{eqnarray} $

(2.10)

where $C'_{v}=\sup_{|z|<1}v(z)C_{v}\max\big\{\frac{1}{v(z)}, \frac{1}{v(p)}\big\}\log\frac{1+\rho (z, p)}{1-\rho (z, p)}$ is a positive constant depends only on $v$. Hence, by the definition of $C_{[n]}$ in (1.1),

$ \|C_{[n]}-C_{p}\|_{v}=\sup\limits_{\|f\|_{v}=1} \|C_{[n]}f(z)-C_{p}f(z)\|_{v}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{C'_{v}\sum_{k=1}^{n}\sup_{\|f\|_{v}=1}\|C_{\varphi^{[k]}}f(z)-C_{p}f(z)\|_{v}}{n}. $

By (2.10), we have

$ \begin{eqnarray*}\|C_{[n]}-C_{p}\|_{v}\leq \frac{1-\|\varphi\|^{n}_{\infty}}{n(1-\|\varphi\|_{\infty})}, \end{eqnarray*} $

thus (2.8) follows, proving Theorem 2.1.

The classical Hardy spaces $H^{p}\ (0< p<\infty)$ over the unit disc $\mathbb{U}=\{z\in \mathbb{C}:|z|< 1\}$ is the collection of functions analytic on the unit disc $\mathbb{U}$, satisfying

$ \|f\|_{p}=(\sup\limits_{0<r<1}\int_{\mathbb{T}}|f(r\xi)|^{p}\rm d \it m(\xi))^{1/p}<\infty, $

where $\mathbb{T}$ is the unit circle $|z|=1$, $m$ is the normalized arc-length measure on $\mathbb{T}$.

If $\varphi$ is a function holomorphic on $\mathbb{U}$ with $\varphi(\mathbb{U})\subset \mathbb{U}$, then $\varphi$ induces a linear composition operator $C_{\varphi}$ on the space Hol$(\mathbb{U})$ of all functions holomorphic on $\mathbb{U}$ (see [3]) as follows

$ C_{\varphi}f=f\circ \varphi \quad(f \in {\rm Hol} (\mathbb{U})). $

Various prospects of such operators were extensively studied (see [3]). In this section, we will examine mean ergodicity of composition operator in Hardy space of Hol$(\mathbb{U})$. The main result of this section is as follows:

Theorem 3.1  Let $\varphi \in S $ be a non-inner function. If for some $w\in \mathbb{U}, \ \varphi(w)=w$, then $C_{\varphi}$ is mean ergodic in $H^{p}\ (0< p<\infty)$.

Proof   From the proof of Proposition 2.4 in [5] and Theorem 3.1 in [6], we know that only the case $p=2$ needs to be verified. We will follow the proof of Theorem 1 in [8].

If $w=0$, denote $H_{0}^{2}=\{f\in H^{2}\}$, recall that $\|C_{\varphi}|H_{0}^{2}\|=\delta<1$ (see [4]). For any $f\in H^{2}, \ \|f\|_{2}=1$,

$ \begin{eqnarray*}\|C_{\varphi}f-C_{0}f\|_{2}=\|C_{\varphi}(f-f(0))\|_{2} \leq\delta\|f-f(0)\|_{2}, \end{eqnarray*} $

thus,

$ \begin{eqnarray*}\|C^{n}_{\varphi}f-C_{0}f\|_{2}=\|C_{\varphi}(f\circ \varphi^{[n-1]}-f(0))\|_{2} \leq\delta\|f\circ \varphi^{[n-1]}-f(0)\|_{2}. \end{eqnarray*} $

Iterating yields

$ \begin{eqnarray}\label{esi c norm}\|C^{n}_{\varphi}-C_{0}\|\leq\delta^{n}.\end{eqnarray} $

(3.11)

Recall Cesàro means in (1.1), combining with (3.11), we have

$ \begin{eqnarray*}\|C_{[n]}-C_{0}\| =\big\|\frac{1}{n}\sum\limits_{m=1}^{n}C^{m}_{\varphi}-C_{0}\big\|\leq\frac{\sum_{m=1}^{n}\|C^{m}_{\varphi}-C_{0}\|}{n}\leq\frac{\delta-\delta^{n}}{n(1-\delta)}, \end{eqnarray*} $

thus the Cesàro means of $C_{\varphi}$ converges and Theorem 2.1 is proved in this case.

Consider the selfinverse conformal automorphism $\alpha_{w}(z)=(w-z)/(1-\overline{w}z)$ and set $\psi=\alpha_{w}\circ\varphi\circ\alpha_{w}$. It is obvious that $\psi(0)=0$. Thus, $\|C^{n}_{\psi}-C^{m}_{\psi}\|\rightarrow 0$ if $m, n \rightarrow\infty$. For each $k$, we have $C^{k}_{\varphi}= C_{\alpha_{w}}C^{k}_{\psi}C_{\alpha_{w}}$ such that $ \|C^{n}_{\varphi}-C^{m}_{\varphi}\|\leq\|C_{\alpha_{w}}\|^{2} \|C^{n}_{\psi}-C^{m}_{\psi}\|\rightarrow 0$ if $m, n \rightarrow\infty$. This shows that $\{C^{n}_{\varphi}\}$ is norm-convergent. From Theorem 3.1 in [6], $C_{\varphi^{[n]}}=\varphi^{[n]}\rightarrow w$ converges weakly in $H^{2}$. Since the set of all composition operators is weakly sequentially compact (see [9] or Remark 1 in [8]), we conclude that $\|C^{n}_{\varphi}-C_{w}\|\rightarrow0.$ From (3.11) and since $\alpha_{w}(z)$ is a selfinverse conformal automorphism, we have

$ \|C_{[n]}-C_{w}\| \;=\;\big\|\frac{1}{n}\sum\limits_{m=1}^{n}C^{m}_{\varphi}-C_{w}\big\|\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\;\frac{\sum_{m=1}^{n}\|C_{\alpha_{w}}C^{m}_{\psi}C_{\alpha_{w}}- C_{w}\|}{n}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\; \frac{\|C_{\alpha_{w}}\|^{2}\sum_{m=1}^{n}\|C^{m}_{\psi}- C_{\alpha_{w}}C_{w}C_{\alpha_{w}}\|}{n}\\\;\;\;\;\;\;\;\;\;\;\;\;\;=\; \frac{\|C_{\alpha_{w}}\|^{2}\sum_{m=1}^{n}\|C^{m}_{\psi}- C_{0}\|}{n}\leq\frac{\delta-\delta^{n}}{n(1-\delta)}, $

thus the Cesàro means of $C_{\varphi}$ converges and Theorem 2.1 is proved.

The space $H^{2}_{d, \beta}$ (see [7]) over the unit ball $\mathbb{B_{N}}= \{z=(z_{1}, z_{2}, \cdots, z_{N} )\in \mathbb{C}^{N}:|z|=(\sum\limits_{k=1}^{N}|z_{k}|^{2})^{1/2}< 1\}$ is the collection of functions analytic on the unit ball $\mathbb{B}^{N}$ with reproducing kernel

$ k_{\beta}(z, w)=\frac{1}{(1-\langle z, w\rangle)^{\beta}}, $

where $\mathbb{T}^{N}$ is the unit circle $|z|=1$, $m$ is the normalized arc-length measure on $\mathbb{T}^{N}$.

The Schur-Agler class is the set of all holomorphic mapping $\varphi:\mathbb{B_{N}}\rightarrow \mathbb{B_{N}} $ for which the Hermitian kernel,

$ k^{\varphi}(z, w)=\frac{1-\langle\varphi(z), \varphi(w)\rangle}{(1-\langle z, w\rangle)^{\beta}} $

is positive semidefinite (see [7]).

A holomorphic self-map $\varphi$ of $\mathbb{B}^{N}$ will be called hyperbolic if $\varphi$ has no fixed point and dilation coefficient $\alpha <1$ (see [7]). The main result of this section is as follows:

Theorem 3.2  If $\varphi $ is a hyperbolic self-map of $\mathbb{B}^{N}$ in the Schur-Agler class, then $C_{\varphi}$ is not mean ergodic in $H^{2}_{d, \beta}$.

Proof  Our proof is an argument by contradiction. If $C_{\varphi}$ were mean ergodic in $H^{2}_{d, \beta}$, by Corollary 1.5 in [7], we have the following

$ \begin{eqnarray}\label{esi Jury ite norm} \|C^{n}_{\varphi}\|\geq A (1-|\varphi_{n}(0)|)^{-\beta/2}, \end{eqnarray} $

(3.12)

where $A$ is some positive constant independent of $n$. If $\varphi $ is a hyperbolic self-map of $\mathbb{B}^{N}$, we have

$ \begin{eqnarray}\label{est Jura fai n} \lim\limits_{n\rightarrow\infty}(1-|\varphi_{n}(0)|^{2})^{1/n}=\alpha\end{eqnarray} $

(3.13)

from Theorem 3.5 in [7]. Recall Cesàro means in (1.1) and (1.2), combining with (3.12) and (3.13), for some fixed $\varepsilon_{0}>0$ satisfying

$ \begin{eqnarray}\label{est vary 0} \alpha+\varepsilon_{0}<1, \end{eqnarray} $

(3.14)

there exists some $M>0$, if $n>M$, we have

$ \begin{eqnarray*}\big\|C_{[n]}-\frac{(n-1)}{n}C_{[n-1]}\big\| =\frac{1}{n}\big \|C^{n}_{\varphi}\big\|\geq A\frac{(1/ (\alpha+\varepsilon_{0}))^{n\beta/2}}{n} .\end{eqnarray*} $

Thus, by (3.14), let $n\rightarrow\infty$, we have $\big\|C_{[n]}-\frac{(n-1)}{n}C_{[n-1]}\big\|\rightarrow\infty$ which contradicts to the norm convergence of $C_{[n]}$, proving Theorem 3.1.