Characterizing Fréchet-Schwartz spaces via power bounded operators (Q2928193)

The authors characterize Köthe echelon spaces which are Schwartz in terms of the convergence of the Cesàro means of power bounded operators acting on those spaces. The authors introduce the notion of a rapidly convergent and rapidly mean ergodic sequence of operators. This notion plays a crucial rôle in their investigation. The main results (Theorem 5.6 and Proposition 5.8) prove that a Köthe echelon space \(\lambda_p(A)\) \((1<p<+\infty)\) is Schwartz if and only if every power bounded operator on \(\lambda_p(A)\) is rapidly mean ergodic. The paper complements the earlier work of the authors on the subject of mean ergodicity of Fréchet (and, more generally, barrelled) spaces.