ANALYTIC CORE OF OPERATORS

Abstract. This paper concerns a localized version of the single valued extension property of a bounded operator T E L(X), where X is a Banach space, at a point AO E C. We shall relate this property to the ascent and the descent of AoI- T, as well as to some spectral subspaces as the quasi-nilpotent part and the analytic core of AoI- 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 £p(N), where 1 <:: p < oc. 1. The single-valued extension property One of basic properties in local spectral theory is the so-called single valued extension property for bounded operators on Banach spaces. This property is enjoyed by several classes of operators as the decomposable operators, as well as other classes of operators; we refer to the excellent monograph by Laursen and Neumann [16] for a modern treatment of the theory of decomposable operators. In this paper we shall consider the following local version of this property, introduced by Finch [13] and studied later by several authors [18, 19, 26, 1, 2, 3