On generalized a-Browder's theorem
We characterize the bounded linear operators T satisfying generalized a-Browder's theorem, or generalized a-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H0(�I T) asbelongs to certain sets of C. In the last part we give a general framework in which generalized a-Weyl's theorem follows for several classes of operators. 1. Preliminaries. Let L(X) denote the space of bounded linear oper- ators on an infinite-dimensional complex Banach space X. For T ∈ L(X), denote by α(T) the dimension of the kernel ker T, and by β(T) the codi- mension of the range T(X). The operator T ∈ L(X) is called upper semi- Fredholm if α(T) < ∞ and T(X) is closed, and lower semi-Fredholm if β(T) < ∞. If T is either upper or lower semi-Fredholm then it is said to be semi-Fredholm; finally, T is a Fredholm operator if it is both upper and lower semi-Fredholm. If T ∈ L(X) is semi-Fredholm, then its index is defined by ind T := α(T) − β(T). For every T ∈ L(X) and a nonnegative integer n we shall denote by T(n) the restriction of T to T n (X) viewed as a map from T n (X) into itself (we set T(0) = T). Following Berkani ((6), (11) and (8)), T ∈ L(X) is said to be semi B-Fredholm (resp., B-Fredholm, upper semi B-Fredholm, lower semi B-Fredholm) if for some integer n ≥ 0 the range T n (X) is closed and T(n) is a semi-Fredholm (resp., Fredholm, upper semi-Fredholm, lower semi- Fredholm) operator. Note that in this case T(m) is semi-Fredholm for all m ≥n ((11)). This enables one to define the index of a semi B-Fredholm op- erator as ind T = ind T(n). The class of all upper semi B-Fredholm operators 2000 Mathematics Subject Classification: Primary 47A10, 47A11; Secondary 47A53, 47A55.