Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems (original) (raw)

2008, Integral Equations and Operator Theory

A Banach space operator T ∈ B(X) is polaroid if points λ ∈ 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 ∈ B(X), let M C denote the operator matrix A C 0 B. If A is polaroid on π 0 (M C) = {λ ∈ isoσ(M C) : 0 < dim(M C − λ) −1 (0) < ∞}, M 0 satisfies Weyl's theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points λ ∈ σ w (M 0) \ σ SF+ (A) and B has SVEP at points µ ∈ σ w (M 0) \ σ SF− (B), or, (ii) both A and A * have SVEP at points λ ∈ σ w (M 0) \ σ SF+ (A), or, (iii) A * has SVEP at points λ ∈ σ w (M 0)\σ SF+ (A) and B * has SVEP at points µ ∈ σ w (M 0)\σ SF− (B), then σ(M C) \ σ w (M C) = π 0 (M C). Here the hypothesis that λ ∈ π 0 (M C) are poles of the resolvent of A can not be replaced by the hypothesis λ ∈ π 0 (A) are poles of the resolvent of A. For an operator T ∈ B(X), let π a 0 (T) = {λ : λ ∈ isoσ a (T), 0 < dim(T − λ) −1 (0) < ∞}. We prove that if A * and B * have SVEP, A is polaroid on π a 0 (M C) and B is polaroid on π a 0 (B) , then σ a (M C) \ σ aw (M C) = π a 0 (M C).