Mean ergodic composition operators on \(H^{\infty} (\mathbb{B}_n)\) (Q2121631)

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,for given \(T\in L(X)\), its \textit{Cesàro means} are defined by \[ T_{[n]}:=\frac{1}{n}\sum_{m=1}^{n}T^{m}, \ \ n\in \mathbb{N}, \] from which one could routinely verify \[ \frac{1}{n}T^{n}=T_{[n]}-\frac{(n-1)}{n}T_{[n-1]}, \ \ n\in \mathbb{N},\tag{1} \] where \(T_{[0]}\) is the identity operator on \(X\). An operator \(T\) is \textit{mean ergodic} if \(\{T_{[n]}\}_{n=0}^{\infty}\) is a convergent sequence in \(L_{s}(X)\). The space \(H^{\infty}(\mathbb{B_{N}})\) (over the unit ball \(\mathbb{B_{N}}= \{z=(z_{1}, z_{2}, \dots, z_{N} )\in \mathbb{C}^{N}:|z|=(\sum_{k=1}^{N}|z_{k}|^{2})^{1/2}< 1\}\) is the collection of functions analytic on the unit ball \(\mathbb{B}^{N}\)endowed with the norm \[ \|f\|_{\infty}=\sup_{|z|<1}|f(z)|<\infty. \] Let \(S\) denote the subset of \(H^{\infty}\) consisting of the analytic selfmaps of \(\mathbb{U}\). If \(\varphi\) is a function holomorphic on \(\mathbb{B}^{N}\) with \(\varphi(\mathbb{B}^{N})\subset \mathbb{B}^{N}\), then \(\varphi\) induces a linear \textit{composition operator} \(C_{\varphi}\) on the space \(\mathrm{Hol}(\mathbb{B}^{N})\) of all functions holomorphic on \(\mathbb{B}^{N}\) as follows \[ C_{\varphi}f=f\circ \varphi \quad(f \in \mathrm{Hol}(\mathbb{B}^{N})). \] In this paper, the author studies mean ergodic composition operators on \(H^{\infty}(\mathbb{B_{N}})\). Mean ergodic properties of composition operators are characterized in terms of spectra and fixed points. The author also proves the equivalence of mean ergodic properties and uniformly mean ergodic properties for such operators. The results can be regarded as natural generalization from the existing ones of unit disk.