New Browder-Weyl type theorems for direct sum

Abstract

In this article, we study the stability of new Browder-Weyl type theorems for orthogonal direct sum $S\oplus T,$ where $S$ and $T$ are bounded linear operators acting on Banach spaces. We characterize preservation of properties
$(W {\scriptstyle \Pi})$ and$ (UW {\scriptstyle \Pi_a})$ under direct sum $S\oplus T.$ Futhermore, we show if $S$ and $T$ satisfy property $(W {\scriptstyle E})$ (resp. $(UW {\scriptstyle E_a})$ ) under certain conditions, then $S\oplus T$ satisfies property $(W {\scriptstyle E})$ (resp. $(UW {\scriptstyle E_a})$ ) if and only if it satisfies generalized Weyl's theorem (resp. generalized a-Weyl's theorem ).