Generalized Weyl\u2019s theorem and property (gw) for upper triangular operator matrices

Abstract

It is known that if Ain mathscr {L}(mathscr {X}) and Bin mathscr {L}(mathscr {Y}) are Banach operators with the single-valued extension property, SVEP, then the matrix operator M_mathrm{{C}}=begin{pmatrix} A &{} C 0&{} B end{pmatrix} has SVEP for every operator Cin mathscr {L}(mathscr {Y},mathscr {X}), and hence obeys generalized Browder’s theorem. This paper considers conditions on operators A, B, and M_0 ensuring generalized Weyl’s theorem and property (Bw) for operators M_mathrm{{C}}. Moreover, certain conditions are explored on Banach space operators T and S so that Toplus S obeys property (gw).