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 ∈ H(σ(T )). Also, we give necessary and sufficient 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 infinite-dimensional complex Banach space, L(X) the algebra of all bounded linear operators on X and K(X) its ideal of compact operators. For an operator T ∈ L(X), write T * for its adjoint; N (T ) for its null space; R(T ) for its range; σ(T ) for its spectrum; σ su (T ) for its surjective spectrum; σ ap (T ) for its approximate spectrum; and σ p (T ) for its point spectrum. From [29] we recall that for T ∈ L(X), the ascent a(T ) and the descent d(T ) are given by a(T ) = inf{n}, respectively; the infimum over the empty set is taken to be ∞. If the ascent and descent of T ∈ L(X) are both finite, then a(is closed and either dim N (T ) or codim R(T ) is finite. For such an operator the index is defined by ind(T ) = dim N (T ) − codim R(T ), and if the index is finite, T is said to be Fredholm. Also, an operator T ∈ L(X) is said to be Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of finite ascent and descent. For T ∈ L(X), the essential spectrum σ e (T ), the Weyl spectrum σ w (T ) and the Browder spectrum σ b (T ) are defined by

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Abstract. For a bounded operator T acting on a complex Banach space, we show that if T − λ is not surjective, then λ is an isolated point of the surjective spectrum σ su (T ) of T if and only if X = H 0 (T −λ)+K (T −λ), where H 0 (T ) is the quasinilpotent part of T and K(T ) is the analytic core for T . Moreover, we study the operators for which dim K(T ) < ∞. We show that for each of these operators T , there exists a finite set E consisting of Riesz points for T such that 0 ∈ σ(T ) \ E and σ(T ) \ E is connected, and derive some consequences.

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In the present paper we examine the stability of Weyl's theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl's theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that F n is finite rank, then T + F obeys Weyl's theorem. Further, we establish that if T is finite-isoloid, then Weyl's theorem is transmitted from T to T + R, for every Riesz operator R commuting with T . Also, we consider an important class of operators that satisfy Weyl's theorem, and we give a more general perturbation results for this class. (2000). 47A53, 47A55 and 46B04. Mathematics Subject Classification