Ascent, Descent, Quasi-nilpotent Part and Analytic Core of Operators

Pietro Aiena, Maria_Teresa Biondi

This paper concerns a localized version of the single valued extension property of a bounded operator $T\in L(X)$, where $X$ is a Banach space, at a point $\lambda_0 \in \Bbb C$. 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 $1\leq p< \infty$.