Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

A Banach space operator $T\in B(\mathcal{X})$ is polaroid if points $\lambda \in \text{iso}\sigma \sigma (T)$ 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 ${\pi }_0(M_C)=\{\lambda \in \text{iso}\sigma (M_C)0<\dim (M_C-\lambda {)}^{-1}(0)<\mathrm{\infty }\}$, $M_0$ 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 $T\in B(\text{X})$, let $\pi_0^a(T)={\lambda:\lambda\in\iso\sigma_a(T), 0<\dim(T-\lambda)^{-1}(0)<\infty}$\mathrm{.}Weprovethatif$A^$and$B^$ have SVEP, $A$ is polaroid on ${\pi }_0^a(\text{M})$ and $B$ is polaroid on ${\pi }_0^a(B)$, then ${\sigma }_a(\text{M})\setminus {\sigma }_{aw}(\text{M})={\pi }_0^a(\text{M})$.