Growth Conditions for Operators With Smallest Spectrum

Abstract

Let A be an invertible operator on a complex Banach space X. For a given alpha >= 0, we define the class D-A(alpha) (Z) (resp. D-A(alpha) (Z(+))) of all bounded linear operators T on X for which there exists a constant C-T > 0, such that parallel to A(n)TA(-n)parallel to <= C-T ( 1 + vertical bar n vertical bar)(alpha), for all n is an element of Z ( resp. n is an element of Z(+)). We present a complete description of the class D-A(alpha) (Z) in the case when the spectrum of A is real or is a singleton. If T is an element of D-A (Z) (= D-A(0) (Z)), some estimates for the norm of AT - TA are obtained. Some results for the class D-A(alpha) (Z(+)) are also given.