Local spectrum, local spectral radius, and growth conditions

Abstract

Let X be a complex Banach space and \(x\in X.\) Assume that a bounded linear operator T on X satisfies the condition $$\begin{aligned} \left\| e^{tT}x\right\| \le C_{x}\left( 1+\left| t\right| \right) ^{\alpha }\quad \left( \alpha \ge 0\right) , \end{aligned}$$for all \(t\in {\mathbb {R}} \) and for some constant \(C_{x}>0.\) For the function f from the Beurling algebra \(L_{\omega }^{1}\left( {\mathbb {R}} \right) \) with the weight \(\omega \left( t\right) =\left( 1+\left| t\right| \right) ^{\alpha },\) 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 \(0\le \alpha <1,\) 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.