Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

A Banach space operator $Tin B({cal X})$ is polaroid if points $lambdainisosigmasigma(T)$ are poles of the resolvent of $T$. Let $sigma_a(T)$, $sigma_w(T)$, $sigma_{aw}(T)$, $sigma_{SF_+}(T)$ and $sigma_{SF_-}(T)$ 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 $Cin B({cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $pi_0(M_C)={lambdainisosigma(M_C) 0<dim(M_C-lambda)^{-1}(0)<infty}$, $M_0$ satisfies Weyls theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B^*$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, then $sigma(M_C)setminussigma_w(M_C)=pi_0(M_C)$. Here the hypothesis that $lambdainpi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $lambdainpi_0(A)$ are poles of the resolvent of $A$. For an operator $Tin B(X)$, let $pi_0^a(T)={lambda:lambdainisosigma_a(T), 0<dim(T-lambda)^{-1}(0)<infty}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $pi_0^a(M)$ and $B$ is polaroid on $pi_0^a(B)$, then $sigma_a(M)setminussigma_{aw}(M)=pi_0^a(M)$.