Weyl's and Browder's theorems for operators satisfying the SVEP

Abstract

We study Weyl's and Browder's theorem for an operator T on a Banach space such that T or its adjoint has the single-valued extension property. We establish the spectral mapping theorem for the Weyl spectrum, and we show that Browder's theorem holds for f(T ) for every f 2H( (T )). Also, we give necessary and sucien t conditions for such T to obey Weyl's theorem. Weyl's theorem in an important class of Banach space operators is also studied. 1. Introduction. Throughout this paper, X denotes an innite-dimen- sional complex Banach space, L(X) the algebra of all bounded linear op- erators on X and K(X) its ideal of compact operators. For an operator T 2 L(X), write T for its adjoint; N(T ) for its null space; R(T ) for its range; (T ) for its spectrum; su(T ) for its surjective spectrum; ap(T ) for its approximate spectrum; and p(T ) for its point spectrum. From (29) we recall that for T 2L(X), the ascent a(T ) and the descent d(T ) are given by a(T ) = inffn 0 : N(T ) n = N(T ) n+1 g and d(T ) = inffn 0 : R(T ) n = R(T ) n+1 g, respectively; the inm um over the empty set is taken to be1. If the ascent and descent of T 2L(X) are both nite, then a(T ) = d(T ) = p, X = N(T ) p R(T ) p and R(T ) p is closed. An operator T 2 L(X) is called semi-Fredholm if R(T ) is closed and either dimN(T ) or codimR(T ) is nite. For such an operator the index is dened by ind(T ) = dimN(T ) codimR(T ), and if the index is nite, T is said to be Fredholm. Also, an operator T 2L(X) is said to be Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of nite ascent and descent. For T 2L(X), the essential spectrum e(T ), the Weyl spectrum w(T ) and the Browder spectrum b(T ) are dened by e(T ) =f 2C : T is not Fredholmg; w(T ) =f 2C : T is not Weylg;