Generalizations of Cesàro means and poles of the resolvent
Volume 164 / 2004
Studia Mathematica 164 (2004), 257-281
MSC: Primary 47A35, 47A10.
DOI: 10.4064/sm164-3-5
Abstract
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 $A$.
