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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 AL(X), BL(Y) and CL(X,Y), denote by M C the operator defined on XY by AC0B.

In this article, the defect set D Σ =(Σ(A)Σ(B))Σ(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 Σ =) and the classes of operators C for which stability holds of M C using local spectral theory.

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