Local spectrum, local spectral radius, and growth conditions.

Let X be a complex Banach space and x ∈ X. Assume that a bounded linear operator T on X satisfies the condition e tT x ≤ C x 1 + t α α ≥ 0 , for all t ∈ R and for some constant C x > 0. For the function f from the Beurling algebra L ω 1 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 a 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 Tx via the local spectral radius of T at x are obtained. Some applications of these results are also given. [ABSTRACT FROM AUTHOR]