Spectrum, Resolvent and Power Boundedness

Abstract

Let L be a bounded operator in a complex Banach space X. In this space we denote the norm by ‖⋅‖ and the same notation is used for the induced operator norms. Unless explicitly stated otherwise, continuity, convergence etc. is to be understood in terms of the norm topology in X and in the uniform operator topology for operators. Our main goal to start with is to estimate powers of L in the form

$$||L^k|| \leq Cr^k, \text{ for all}\ \ k=0,1,2, ...$$

((2.1.1))

.