Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

Abstract

A Banach space operator $T\in B(\mathcal{X})$ is polaroid if points $λ\in\isoσσ(T)$ are poles of the resolvent of $T$. Let ${\sigma }_a(T)$, ${\sigma }_w(T)$, ${\sigma }_{aw}(T)$, ${\sigma }_{S{F}_{+}}(T)$ and ${\sigma }_{S{F}_{-}}(T)$ 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 $C\in B(\mathcal{X})$, 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 $\lambda \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$ and $B$ has SVEP at points $\mu \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{-}}(B)$, or, (ii) both $A$ and $A^{\ast }$ have SVEP at points $\lambda \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$, or, (iii) $A^{\ast }$ has SVEP at points $\lambda \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$ and $B^{\ast }$ has SVEP at points $\mu \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{-}}(B)$, then $\sigma (M_C)\setminus {\sigma }_w(M_C)={\pi }_0(M_C)$. Here the hypothesis that $\lambda \in {\pi }_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $\lambda \in {\pi }_0(A)$ 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 $A^{\ast }$ and $B^{\ast }$ have SVEP, $A$ is polaroid on $π_0^a(\M)$ and $B$ is polaroid on ${\pi }_0^a(B)$, then $σ_a(\M)\setminusσ_{aw}(\M)=π_0^a(\M)$.

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