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 ) {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 ∈ B ( X ) T \in B(X) has the property ( β ∗ ) ({\beta ^{\ast }}) introduced by Fong, then X T ∗ ∗ ( F ) = X T ( C ∖ F ) ⊥ X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot } for every closed set F F in the complex plane C \mathbb {C} .