Elements with generalized bounded conjugation orbits

Drissi, Driss; Mbekhta, Mostafa

doi:10.1090/S0002-9939-01-05945-7

Elements with generalized bounded conjugation orbits
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by Driss Drissi and Mostafa Mbekhta PDF

Proc. Amer. Math. Soc. 129 (2001), 2011-2016 Request permission

Abstract:

For a pair of linear bounded operators $T$ and $A$ on a complex Banach space $X$, if $T$ commutes with $A,$ then the orbits $\{A^n TA^{-n}\}$ of $T$ under $A$ 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.

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