Quasi-reflexive Fréchet spaces and mean ergodicity
Several results showing the relation between the properties of a Banach space \(X\) with a Schauder basis and the existence of power bounded non mean ergodic operators on \(X\) were proved in the important paper by \textit{V.\,P.\thinspace Fonf, M.\,Lin} and \textit{P.\,Wojtaszczyk} [J.~Funct.\ Anal.\ 187, No.\,1, 146--162 (2001; Zbl 1017.46007)]. The proof of their main results is based on classical results of Pełczyński and of Zippin, connecting bases with reflexivity. Their work was continued in the setting of Fréchet spaces by \textit{A.\,A.\thinspace Albanese, W.\,J.\thinspace Ricker} and the reviewer [Ann.\ Acad.\ Sci.\ Fenn., Math.\ 34, No.\,2, 401--436 (2009; Zbl 1194.47012)]. The results of Pełczyński and of Zippin were there extended to Fréchet spaces, thus solving an open problem of Kalton. The aim of the present paper is to complement the results of the aforementioned papers by obtaining a characterization of nonreflexive Fréchet spaces \(E\) with a basis which are quasi-reflexive of order one (i.e., \(E\) has codimension one in its bidual \(E''\)) in terms of the existence of power bounded operators which are not mean ergodic in the space \(E\) or in the dual \(E'\). To do this, the author shows that certain results concerning bases in Banach spaces due to Singer in 1963 can be extended to Fréchet spaces. This extension requires new ideas and techniques.
scientific article; zbMATH DE number 5625400
scientific article