Power bounded weighted composition operators (Q416790)

A bounded operator \(T:X\to X\) is said to be power bounded if \(\sup_n\|T^n\|<\infty\) and uniformly mean ergodic whenever \(\frac{1}{n}\sum_{k=1}^n T^k\) is convergent. The author studies these properties for weighted composition operators, meaning \(C_{\phi,\psi}(f)=\psi (f\circ\phi)\) for given \(\phi,\psi\in H(\mathbb D)\) and \(\|\phi\|_\infty\leq 1\), acting on spaces \(H^\infty_v\) defined by \(\sup_{|z|<1}v(z)|f(z)|<\infty\) for a given \(v:\mathbb D\to (0,\infty)\) continuous and bounded. Previous results in this direction are known. In the case \(v=1\) and \(\phi(z)=z\), that is, multiplication operators acting on \(H^\infty\), they were given by \textit{J. Bonet} and \textit{W. J. Ricker} [``Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions'', Arch. Math. 92, No. 5, 428--437 (2009; Zbl 1216.47014)] and in the case \(\psi(z)=1\) on \(H^\infty_v\), that is, composition operators acting on \(H_v^\infty\), were given by the author [``Power bounded composition operators'', Comput. Methods Funct. Theory 12, No. 1, 105--117 (2012; Zbl 1252.47029)].NEWLINENEWLINE The main results of the paper under review establish that \(C_{\phi,\psi}\) are power bounded if and only if they are similar to contractions under some assumptions on \(\phi\) and \(v\). The author also finds some sufficient conditions on \(\phi,\psi\) and \(v\) to get that \(C_{\phi,\psi}\) are uniformly mean ergodic.