UPPER TRIANGULAR OPERATORS WITH SVEP
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A Banach space operator A ∈ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A ∈ (HP), if every part of A is polaroid. Let [수식]Xi, where Xi are Banach spaces, and let A denote the class of upper triangular operators A = (Aij )1≤i,j·n, Aij ∈ B(Xj ,Xi) and Aij = 0 for i > j. We prove that operators A ∈ A such that Aii for all 1≤ i · n, and A¤ have the single–valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A 2 A such that Aii ∈ (HP) for all 1 ≤ i · n are polaroid and have SVEP; hence they satisfy Weyl’s theorem. Furthermore, A + R satisfies Browder’s theorem for all upper triangular operators R, such that [수식] is a Riesz operator, which commutes with A.
Abstract
A Banach space operator A ∈ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A ∈ (HP), if every part of A is polaroid. Let [수식]Xi, where Xi are Banach spaces, and let A denote the class of upper triangular operators A = (Aij )1≤i,j·n, Aij ∈ B(Xj ,Xi) and Aij = 0 for i > j. We prove that operators A ∈ A such that Aii for all 1≤ i · n, and A¤ have the single–valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A 2 A such that Aii ∈ (HP) for all 1 ≤ i · n are polaroid and have SVEP; hence they satisfy Weyl’s theorem. Furthermore, A + R satisfies Browder’s theorem for all upper triangular operators R, such that [수식] is a Riesz operator, which commutes with A.
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