EXTENDED WEYL-TYPE THEOREMS FOR DIRECT SUMS

In this paper, we study the stability of extended Weyl and Browdertype theorems for orthogonal direct sum S circle plus 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 Pi(0)(a) (T) subset of sigma(a) (S) and Pi(0)(a) (S) subset of sigma(a) (T), we characterize preservation of property (ab) under direct sum S circle plus T. Furthermore, we show that if S and T satisfy generalized a-Browder's theorem, then S circle plus T satisfies generalized a-Browder's theorem if and only if sigma(SBF)-(+) (S circle plus T)= sigma(SBF)-(+) (S)boolean OR sigma(SBF)-(+) (T); which improves a recent result of [13] by removing certain extra assumptions.