Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

TL;DR

This paper investigates the spectral properties of upper triangular operator matrices, establishing conditions under which Weyl's theorem and related spectral equalities hold, especially involving SVEP, polaroid operators, and Browder's theorems.

Contribution

It provides new spectral characterizations for operator matrices using SVEP, polaroid conditions, and Browder-type theorems, extending previous results in operator theory.

Findings

Under certain SVEP and polaroid conditions, the spectrum of the operator matrix equals its Weyl spectrum.

Proves that $\sigma(M_C)ackslash \sigma_w(M_C) = \pi_0(M_C)$ under specified hypotheses.

Establishes spectral equalities for approximate spectra when dual operators have SVEP.

Abstract

A Banach space operator $T\in B({\cal X})$ is polaroid if points $\lambda\in\iso\sigma\sigma(T)$ 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 $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $\pi_0(M_C)=\{\lambda\in\iso\sigma(M_C) 0<\dim(M_C-\lambda)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B^*$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, 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 $T\in B(\X)$, let $\pi_0^a(T)=\{\lambda:\lambda\in\iso\sigma_a(T), 0<\dim(T-\lambda)^{-1}(0)<\infty\}$. 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 $\sigma_a(\M)\setminus\sigma_{aw}(\M)=\pi_0^a(\M)$.