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Browder spectra and essential spectra of operator matrices. (English) Zbl 1162.47004

From the Introduction: “W. Y. Lee [Proc. Am. Math. Soc. 129, No. 1, 131–138 (2001; Zbl 0965.47011)] has recently considered the filling-in-holes problem of 2×2 upper triangular operator matrices. His main result can be described as follows: if M C =AC0B is an operator acting on the Hilbert space H×K, then

σ w (A)σ w (B)=σ w (M C )W,

where σ w (B·) denotes the Weyl spectrum and W is the union of certain of the holes in σ w (M C ) which happen to be subsets of σ w (A)σ w (B). J. K. Han, H. Y. Lee and W. Y. Lee [Proc. Am. Math. Soc. 128, No. 1, 119–123 (2000; Zbl 0944.47004)] have proved that the above result is also true for spectra of A,B and M c in Banach spaces.

In this note, we show that the passage from σ τ (A)σ τ (B) to σ τ (M c ) is the punching of some open sets in σ τ (A)σ τ (B), where σ τ (·) can be equal to the Browder spectrum or essential spectrum and the result is not true if σ τ (·) is equal to the Kato spectrum, left (right) essential spectrum and left (right) spectrum.”

At the end, the authors present illustrative examples.


MSC:
47A11Local spectral properties
47A10Spectrum and resolvent of linear operators
References:
[1]Lee, W. Y.: Weyl spectra of operator matrices. Proc. Amer. Math. Soc., 129, 131–138 (2001) · Zbl 0965.47011 · doi:10.1090/S0002-9939-00-05846-9
[2]Han, J. K., Lee, H. Y., Lee, W. Y.: Invertible completions of 2 × 2 upper triangular operator matrices. Proc. Amer. Math. Soc., 128, 119–123 (2000) · Zbl 0944.47004 · doi:10.1090/S0002-9939-99-04965-5
[3]Dowson, H. R.: Spectral Theory of Linear Operators, Academic Press, London, New York, 1978
[4]Caradus, S. R.: Generalized Inverses and Operator Theory, Queen’s Paper in Pure and Applied Mathematics, 50, Queen’s Univ., Kingston, 1978
[5]Djordjevic, D. S., Stanimirovic, P. S.: On the generalized Drazin inverse and generalized resolvent. Czechoslovak Math. J., 51(3), 617–634 (2001) · Zbl 1079.47501 · doi:10.1023/A:1013792207970
[6]Djordjevic, S. V., Han, Y. M.: A note of Weyl’s Theorem for operator matrices. Proc. Amer. Math. Soc., 131, 2543–2547 (2002) · Zbl 1041.47006 · doi:10.1090/S0002-9939-02-06808-9
[7]Zhong, H. J.: Structure of Banach Spaces and Operator Ideals, Science Press, Beijing, 2005 (in Chinese)
[8]Djordjevic, D. S.: Perturbations of spectra of operator matrices. J. Operator Theory., 48, 467–486 (2002)
[9]Li, Y., Sun, S. H.: A note on the left essential spectra of operator matrices. Acta Mathematica Sinica, English Series, 23(12), 2235–2240 (2007) · Zbl 1151.47007 · doi:10.1007/s10114-007-0966-0
[10]Berkani, M.: On the equivalence of Weyl and generalized Weyl theorem. Acta Mathematica Sinica, English Series, 23(1), 103–110 (2007) · Zbl 1116.47015 · doi:10.1007/s10114-005-0720-4
[11]Cao, X. H.: A-Browder’s theorem and generalized a-Weyl’s theorem. Acta Mathematica Sinica, English Series, 23(5), 951–960 (2007) · Zbl 1153.47009 · doi:10.1007/s10114-005-0870-4