Generalized Browder’s Theorem and SVEP

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 generalized 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.

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…”

“…Note that in all the papers concerning generalized Browder's theorems (see for instance [7], [15], [12], [8]), there is no trace of the role of B-Browder spectra. Our Corollary 2.10 shows that this is only apparent.…”

On generalized a-Browder's theorem

“…Generalized Browder's theorem for T is equivalent to the SVEP of T at the points λ / ∈ σ bw (T ) [5], and obviously, σ usbf − (T ) ⊆ σ bw (T ). By Theorem 2.4 generalized a-Browder's theorem for T implies generalized Browder's theorem for T .…”

“…Generalized Weyl's theorem may also be described by the quasi-nilpotent part H 0 (λI − T ) as λ ranges over a suitable subset of C. In fact, if we define ∆ 1 (T ) := ∆(T ) ∪ E(T ), in [5] it is shown that generalized Weyl's theorem holds for T if and only if for every λ ∈ ∆ 1 (T ) there exists p := p(λ) ∈ N such that H 0 (λI − T ) = ker (λI − T ) p . Since In an important situation this implication may be reversed: Theorem 2.19.…”

“…By Theorem 1.20 of [5], f (T ) satisfies generalized Weyl's theorem. By [1, Theorem 2.40], f (T * ) = f (T ) * has SVEP.…”