UPPER TRIANGULAR OPERATORS WITH SVEP

Title & Authors
UPPER TRIANGULAR OPERATORS WITH SVEP

Abstract
A Banach space operator A $\small{\in}$ 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 $\small{\in}$ ($\small{\mathcal{H}\mathcal{P}}$), if every part of A is polaroid. Let $\small{X^n\;=\;\oplus^n_{t=i}X_i}$, where $\small{X_i}$ are Banach spaces, and let A denote the class of upper triangular operators A = $\small{(A_{ij})_{1{\leq}i,j{\leq}n}$, $\small{A_{ij}\;{\in}\;B(X_j,X_i)}$ and $\small{A_{ij}}$ = 0 for i > j. We prove that operators A $\small{\in}$ A such that $\small{A_{ii}}$ for all $\small{1{\leq}i{\leq}n}$, and $\small{A^*}$ 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 $\small{\in}$ A such that $\small{A_{ii}}$ $\small{\in}$ ($\small{\mathcal{H}\mathcal{P}}$) for all $\small{1{\leq}i{\leq}n}$ 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 $\small{\oplus^n_{i=1}R_{ii}}$ is a Riesz operator, which commutes with A.
Keywords
Banach space;n-normal operator;hereditarily polaroid operator;single valued extension property;Weyl's theorem;
Language
English
Cited by
1.
A Note on Drazin Invertibility for Upper Triangular Block Operators, Mediterranean Journal of Mathematics, 2013, 10, 3, 1497
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