Local Spectrum, Local Spectral Radius, and Growth Conditions

Abstract

Let X be a complex Banach space and x is an element of X. Assume that a bounded linear operator parallel to etT(x)parallel to <= C-x (">1+vertical bar t vertical bar)(alpha) (alpha >= 0), for all t is an element of R and for some constant Cx > 0. For the function f from the Beurling algebra L omega 1 with the weight omega(t) (>1+t(alpha)) we can define an element in X, denoted by xf, which integrates etTx with respect to f. We present a complete description of the elements xf in the case when the local spectrum of T at x consists of one point. In the case 0 <=alpha<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.