Growth Conditions for Conjugation Orbits of Operators on Banach Spaces

Abstract

Let A be an invertible bounded linear operator on a complex Banach space X. With connection to the Deddens algebras, for a given k is an element of N, we define the class D-A(k) of all bounded linear operators T on X for which the conjugation orbits {A(n)TA(-n)}(n is an element of z) satisfies some growth conditions. We present a complete description of the class D-A(k) in the case when the spectrum of A is positive. Individual versions of Katznelson-Tzafriri theorem and their applications to the Deddens algebras are given. The Hille-Yosida space is used to obtain local quantitative results related to the Katznelson-Tzafriri theorem. Some related problems are also discussed.