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Characterizations of perturbations of spectra of 2×2 upper triangular operator matrices. (English) Zbl 1256.47015

The authors investigate perturbations of the spectrum of a general 2×2 upper triangular operator matrix M C =AC0B acting on the Hilbert space 𝒦, where AB(), BB(𝒦) and CB(𝒦,).

X. H. Cao, M. Z. Guo and B. Meng [Acta Math. Sin., Engl. Ser. 22, No. 1, 169–178 (2006; Zbl 1129.47014)] gave a necessary and sufficient condition for M C to be an upper semi-Fredholm operator (resp., a lower semi-Fredholm operator, a Fredholm operator) and characterized the intersection of the upper semi-Fredholm spectrum and the lower semi-Fredholm spectrum of M C . X. H. Cao and B. Meng [J. Math. Anal. Appl. 304, No. 2, 759–771 (2005; Zbl 1083.47006)] obtained a necessary and sufficient condition for M C to be an upper semi-Weyl operator (resp., a lower semi-Weyl operator, a Weyl operator) and characterized the intersection of the upper semi-Weyl spectrum, the lower semi-Weyl spectrum and the Weyl spectrum of M C . I. S. Hwang and W. Y. Lee [Integral Equations Oper. Theory 39, No. 3, 267–276 (2001; Zbl 0986.47004)] provided a necessary and sufficient condition for M C to be a left invertible operator and characterized the intersection of the left spectrum, the right spectrum and the spectrum of M C .

The authors of the present paper extend all the results mentioned above by the same techniques. Some counterexamples are presented as well.

MSC:
47A55Perturbation theory of linear operators
47A53(Semi-)Fredholm operators; index theories
47A10Spectrum and resolvent of linear operators
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