Ascent, descent, quasi-nilpotent part and analytic core of operators

Abstract

This paper concerns a localized version of the single valued extension property of a bounded operator $Tın L(X)$, where $X$ is a Banach space, at a point $\lambda_0 ın \Bbb C$. We shall relate this property to the ascent and the descent of $\lambda_0 I-T$, as well as to some spectral subspaces as the quasi-nilpotent part and the analytic core of $\lambda_0 I- T$. We shall also describe all these notions in the setting of an abstract shift condition, and in particular for weighted right shift operators on $\ell^p (\Bbb N)$, where $1\leq p< ınfty$.