The single-valued extension property
Harold Garth Dales, Pietro Aiena, Jörg Eschmeier, Kjeld Bagger Laursen, George A. Willis · Cambridge University Press eBooks · 2003
The basic role of the single-valued extension property (SVEP) arises in the spectral decomposition theory, since every decomposable operator T enjoys this property, as does its dual T′ . Indeed, in part IV, Chapter 21 it has been shown that the decomposability of an operator may be viewed as the union of two properties, the so-called Bishop's property ( β ) and the property ( δ ). Property ( β ) for T implies the SVEP for T (see part IV, Chapter 21) and, as observed in part IV, Chapter 23, properties ( β ) and ( δ ) have a complete duality, so that, if T has ( δ ), then the dual T′ has ( β ) and therefore SVEP. The main goal of this chapter is to investigate in detail a localized version of SVEP. First we shall show that local spectral theory provides a suitable frame for some characterizations of the analytical core and of the quasi-nilpotent part. Then we shall use these characterizations to describe the localized SVEP by means of a variety of conditions that involve the analytical core and the quasinilpotent part of an operator, as well as the generalized range and the generalized kernel. The SVEP at a point To explain the role of SVEP in local spectral theory we begin with some preliminary and well-known facts from operator theory.