Uniform mean ergodicity of $C_0$-semigroups\newline in a class of Fréchet spaces

Abstract

Let $(T(t))_{t\geq 0}$ be a strongly continuous $C_0$-semigroup of bounded linear operators on a~Banach space $X$ such that $\lim_{t\to\infty}||T(t)/t||=0$. Characterizations of when $(T(t))_{t\geq 0}$ is uniformly mean ergodic, i.e., of when its Cesàro means $r^{-1}\int_0^r T(s)ds$ converge in operator norm as $r\to\infty$, are known. For instance, this is so if and only if the infinitesimal generator $A$ has closed range in $X$ if and only if $\lim_{\lambda\downarrow 0^+}\lambda R(\lambda, A)$ exists in the operator norm topology (where $R(\lambda,A)$ is the resolvent operator of $A$ at $\lambda$). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of $C_0$-semigroups in particular Fréchet function and sequence spaces are presented.