Abstract
Let \( M_C = \left( {\begin{array}{*{20}c} A & C \\ 0 & B \\ \end{array} } \right) \) be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that
where W is the union of certain of the holes in σ τ (M C ) which happen to be subsets of σ τ (A) ∩ σ τ (B), and σ τ (A), σ τ (B), σ τ (M C ) can be equal to the Browder or essential spectra of A, B, M C , respectively. We also show that the above result isn’t true for the Kato spectrum, left (right) essential spectrum and left (right) spectrum.
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References
Lee, W. Y.: Weyl spectra of operator matrices. Proc. Amer. Math. Soc., 129, 131–138 (2001)
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)
Dowson, H. R.: Spectral Theory of Linear Operators, Academic Press, London, New York, 1978
Caradus, S. R.: Generalized Inverses and Operator Theory, Queen’s Paper in Pure and Applied Mathematics, 50, Queen’s Univ., Kingston, 1978
Djordjevic, D. S., Stanimirovic, P. S.: On the generalized Drazin inverse and generalized resolvent. Czechoslovak Math. J., 51(3), 617–634 (2001)
Djordjevic, S. V., Han, Y. M.: A note of Weyl’s Theorem for operator matrices. Proc. Amer. Math. Soc., 131, 2543–2547 (2002)
Zhong, H. J.: Structure of Banach Spaces and Operator Ideals, Science Press, Beijing, 2005 (in Chinese)
Djordjevic, D. S.: Perturbations of spectra of operator matrices. J. Operator Theory., 48, 467–486 (2002)
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)
Berkani, M.: On the equivalence of Weyl and generalized Weyl theorem. Acta Mathematica Sinica, English Series, 23(1), 103–110 (2007)
Cao, X. H.: A-Browder’s theorem and generalized a-Weyl’s theorem. Acta Mathematica Sinica, English Series, 23(5), 951–960 (2007)
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This work was supported by the National Natural Science Foundation of China (Grants NO.10471025, NO.10771034), the Natural Science Foundation of Fujian Province of China(Grant NO.S0650009), the Education Department Foundation of Fujian Province of China (Grants NO.JA05211, NO.JB06026) and the Foundation of Technology and Development of Fuzhou University(Grant NO.2007-XY-11)
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Zhang, Y.N., Zhong, H.J. & Lin, L.Q. Browder spectra and essential spectra of operator matrices. Acta. Math. Sin.-English Ser. 24, 947–954 (2008). https://doi.org/10.1007/s10114-007-6339-x
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DOI: https://doi.org/10.1007/s10114-007-6339-x