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 $TınL(X)$, where $X$ is a Banach space, at a point ${\lambda }_0ın\mathbb{C}$. 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 ${\ell }^p(\mathbb{N})$, where $1\le p<ınfty$.