Mean ergodic theorems for semigroups of linear operators
In 1938, Yosida proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and T be a linear operator of E into itself such that there exists a constant C with \(\| T^ n\| \leq C\) for \(n=1,2,3,...\), and T is weakly completely continuous, i.e., T maps the closed unit ball of E into a weakly compact subset of E. Then, the Cesàro means \(S_ nx=(1/n)\sum^{n}_{k=1}T^ kx\) converge strongly as \(n\to +\infty\) to a fixed point of T for each \(x\in E.\) In this paper, by using \textit{G. Rodé}'s method [this J. 85, 172-178 (1982; Zbl 0485.47041)], we extend Yosida's theorem to semigroups of linear operators in Banach spaces.
scientific article; zbMATH DE number 3983943