Local spectral mapping theorems
Robert G. Bartle, Constantine A. Kariotis · Bulletin of the American Mathematical Society · 1973
This note is concerned with various "localized versions" of the spectral mapping theorem (cf.[3, VII.3.11]) for bounded linear operators in a complex Banach space.The details of the proof and some additional results will be published elsewhere.Let X be a complex Banach space and let Te B(X\ the Banach algebra of all bounded linear operators on X.We recall (cf.[2, p. 1], [3, p. 1931]) that if T has the single-valued extension property then there exist a maximal open set p T (x) containing p(T) and a unique holomorphic function x T :p T (x) -> X such that (XI -T)x T (À) = x for all Xep T (x).The complementary set a T (x) = C -p T {x\ which we call the local spectrum of x (with respect to T), is compact and is contained in a(T\ the spectrum of T If F £ C is closed we introduce the spectral manifold X T (F)= {xeX:a T (x)^F}.THEOREM 1.Let ƒ be holomorphic on a neighborhood of a(T) and suppose that T and f(T) have the single-valued extension property.Then f{(T T (x)) = (7 /(T) (x) for all x e X.