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.