Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

Abstract

A Banach space operator $T\in B({\cal X})$ is polaroid if points $λ\in\isoσσ(T)$ are poles of the resolvent of $T$. Let $σ_a(T)$, $σ_w(T)$, $σ_{aw}(T)$, $σ_{SF_+}(T)$ and $σ_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $π_0(M_C)=\{λ\in\isoσ(M_C) 0<\dim(M_C-λ)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$ and $B$ has SVEP at points $μ\inσ_w(M_0)\setminusσ_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$ and $B^*$ has SVEP at points $μ\inσ_w(M_0)\setminusσ_{SF_-}(B)$, then $σ(M_C)\setminusσ_w(M_C)=π_0(M_C)$. Here the hypothesis that $λ\inπ_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $λ\inπ_0(A)$ are poles of the resolvent of $A$. For an operator $T\in B(\X)$, let $π_0^a(T)=\{λ:λ\in\isoσ_a(T), 0<\dim(T-λ)^{-1}(0)<\infty\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $π_0^a(\M)$ and $B$ is polaroid on $π_0^a(B)$, then $σ_a(\M)\setminusσ_{aw}(\M)=π_0^a(\M)$.