Elements with generalized bounded conjugation orbits

Abstract

For a pair of linear bounded operators T T and A A on a complex Banach space X X , if T T commutes with A , A, then the orbits { A n T A − n } \{A^n TA^{-n}\} of T T under A A are uniformly bounded. The study of the converse implication was started in the 1970s by J. A. Deddens. In this paper, we present a new approach to this type of question using two localization theorems; one is an operator version of a theorem of tauberian type given by Katznelson-Tzafriri and the second one is on power-bounded operators by Gelfand-Hille. This improves former results of Deddens-Stampfli-Williams.