The Single-Valued Extension Property and Spectral Manifolds

Abstract

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds ${X_T}(F)$ of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if $T \in B(X)$ has the property $({\beta ^{\ast }})$ introduced by Fong, then $X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot }$ for every closed set $F$ in the complex plane $\mathbb {C}$.