Extended Weyl-Type Theorems for Direct Sums

Abstract

In this paper, we study the stability of extended Weyl and Browdertype theorems for orthogonal direct sum S⊕T, where S and T are bounded linear operators acting on Banach space. Two counterexamples shows that property (ab), in general, is not preserved under direct sum. Nonetheless, and under the assumptions that Π0α (T) ⊂σα(S) and Π0α(S) ⊂σα(T), we characterize preservation of property (ab) under direct sum S⊕T. Furthermore, we show that if S and T satisfy generalized a-Browder’s theorem, then S⊕T satisfies generalized a-Browder’s theorem if and only if σSBF-+ (S ⊕T) = σSBF-+(S) ∪σSBF-+(T), which improves a recent result of [13] by removing certain extra assumptions.

Keywords

• property pgabq
• property pgawq
• direct sums
• B-Weyl spectrum