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Perturbation of spectra of operator matrices and local spectral theory. (English) Zbl 1105.47006
Let $X$ and $Y$ be Banach spaces and let $L(X,Y)$ denote the space of all bounded linear operators from $X$ to $Y$ ($L(X):=L(X,X)$). For $A\in L(X)$, $B\in L(Y)$ and $C\in L(X,Y)$, denote by $M_{C}$ the operator defined on $X\oplus Y$ by $\left[\smallmatrix A & C\\ 0 & B \endsmallmatrix \right]$. In this article, the defect set $D^{\Sigma }=(\Sigma (A)\cup \Sigma (B))\setminus \Sigma (M_{C})$ is studied for different spectra including the spectrum, the essential spectrum, the Weyl spectrum and the approximate point spectrum. The obtained results are applied to the stability of such spectra ($D^{\Sigma }=\varnothing$) and the classes of operators $C$ for which stability holds of $M_{C}$ using local spectral theory.

MSC:
47A11Local spectral properties
47A55Perturbation theory of linear operators
47A10Spectrum and resolvent of linear operators
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References:
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