OnThe Local Spectral Properities of Weighted Shift Operator

Abstract

We show, for a class of operators on Banach space, a natural growth condition to guarantee Bishop’s property for weighted shifts, this result leads to a sufficient condition in terms of the underlying weighted shift with property has fat local spectra and approximate point spectra a circle while bilateral weighted shift with property have either fat local spectra or spectrum a circle. We show the tool of the inner local spectral radius. For weighted shift with Dunford’s property (C), both the inner and outer spectral radii turn out to be constant. We consider continuity of local spectra and introduction one generalization of the well- known first resolvent equation by means of local resolvent function. In addition we show several necessary and sufficient conditions for local spectral properties for both unilateral and bilateral weighted shift operators in terms of the weight sequence defining them.