Local Spectrum, Local Spectral radius,and Growth Conditions

Abstract

Let X be a complex Banach space and x ∈ X. Assume that a bounded linear operator T on X satisfies the conditionfor all t ∈ R and for some constant Cx > 0. For the function f from the Beurling algebra L 1 ω (R) with the weight ω (t) = (1 + |t|) α , we can define an element in X, denoted by x f , which integrates e tT x with respect to f. We present complete description of the elements x f in the case when the local spectrum of T at x consists of one-point. In the case 0 ≤ α < 1, some estimates for the norm of T x via local spectral radius of T at x are obtained. Some applications of these results are also given.