WEYL TYPE-THEOREMS FOR DIRECT SUMS

Abstract

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum <TEX>$S{\oplus}T$</TEX>, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property (<TEX>$gb$</TEX>) and if <TEX>${\Pi}(T){\subset}{\sigma}_a(S)$</TEX>, <TEX>${\PI}(S){\subset}{\sigma}_a(T)$</TEX>, then <TEX>$S{\oplus}T$</TEX> possesses property (<TEX>$gb$</TEX>) if and only if <TEX>${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$</TEX>. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then <TEX>$S{\oplus}T$</TEX> satis es generalized Browder's theorem if and only if <TEX>${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$</TEX>.