Generalizations of Cesàro means and poles of the resolvent

Laura Burlando¹

¹ Dipartimento di Matematica dell'Università di Genova Via Dodecaneso 35 16146 Genova, Italy

Studia Mathematica, Tome 164 (2004) no. 3, pp. 257-281

Résumé

An improvement of the generalization—obtained in a previous article [Bu1] by the author—of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators $T$ and $A$ are constructed such that $n^{-2}T^n$ converges uniformly to zero, the sum of the range and the kernel of $1-T$ being closed, and $n^{-3}\sum _{k=0}^ {n-1}A^k$ converges uniformly, the sum of the range of $1-A$ and the kernel of ${(1-A)}^2$ being closed. Nevertheless, $1$ is a pole of the resolvent of neither $T$ nor $A$.

Détail

DOI : 10.4064/sm164-3-5

Keywords: improvement generalization obtained previous article author uniform ergodic theorem poles arbitrary order derived order answer natural questions suggested result examples given namely bounded linear operators constructed converges uniformly zero sum range kernel t being closed sum n converges uniformly sum range a kernel a being closed nevertheless pole resolvent neither nor