Mediterr. j. math. 4 (2007), 215–228 1660-5446/020215-14, DOI 10.1007/s00009-007-0113-2 c  2007 Birkh¨auser Verlag Basel/Switzerland Mediterranean Journal of Mathematics Generalized Browder’s Theorem and SVEP Pietro Aiena and Orlando Garcia Abstract. A bounded operator T ∈ L(X), X a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T , while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying genera- lized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H0(λI -T ) as λ belongs to certain subsets of C. In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators. Mathematics Subject Classification (2000). Primary 47A10, 47A11; Secondary 47A53, 47A55. Keywords. SVEP, Fredholm theory, generalized Weyl’s theorem and general- ized Browder’s theorem. 1. Introduction and definitions Let T ∈ L(X ) be a bounded operator on an infinite-dimensional complex Banach space X and denote by α(T ) the dimension of the kernel ker T , and by β(T ) the codimension of the range T (X ). T ∈ L(X ) is called an upper semi-Fredholm operators if α(T ) < ∞ and T (X ) is closed, while T ∈ L(X ) is called a lower semi-Fredholm operators if β(T ) < ∞. The class of all semi-Fredholm operators is defined by Φ ± (X ) := Φ + (X ) ∪ Φ - (X ), the class of all Fredholm operators is defined by Φ(X ) := Φ + (X ) ∩ Φ - (X ), where Φ + (X ) and Φ - (X ) denote the classes of upper semi-Fredholm operators and lower semi-Fredholm operators, respectively. If T ∈ Φ ± (X ), the index of T is defined by ind (T ) := α(T ) - β(T ). For every T ∈ L(X ) and a nonnegative integer n, by T [n] we shall denote the restriction of T to T n (X ) viewed as a map from the space T n (X ) into itself. According Berkani ([8], [10] and [11]), T ∈ L(X ) is said to be B-Fredholm (resp., upper semi B-Fredholm,