On generalized a-Browder's theorem

Abstract

Abstract. 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 H 0 (λI − T ) as λ belongs 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 operators 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 codimension of the range T (X). The operator T ∈ L(X) is called upper semiFredholm 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

On Drazin invertibility

“…The proof of the equalities (i), (iii) may be found in [6] and [7]. To show the equality (ii), we observe first that…”

Weyl Type Theorems for Left and Right Polaroid Operators

“…Proof. The first assertion has been proved in [9]. The second assertion is clear: for every left polaroid operator we have E a (T ) = Π a 00 (T ).…”

“…To show the first equalities it suffices, by (9), to prove that π a 00 (T ) ⊆ p 00 (T ). Let λ ∈ π a 00 (T ).…”

Drazin spectra of Banach space operators and Banach algebra elements

“…When A = L(X), X a Banach space, the left and the right Drazin spectra of an operator have been introduced, see Definition 2.5 of [7] and Definition 1.1 of [3,4], see also [1,2]. Before recalling these notions, consider the sets…”