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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 ${M}_{C}=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$ is an operator acting on the Hilbert space $H×K$, then

${\sigma }_{w}\left(A\right)\cup {\sigma }_{w}\left(B\right)={\sigma }_{w}\left({M}_{C}\right)\cup W,$

where ${\sigma }_{w}\left(B·\right)$ denotes the Weyl spectrum and $W$ is the union of certain of the holes in ${\sigma }_{w}\left({M}_{C}\right)$ which happen to be subsets of ${\sigma }_{w}\left(A\right)\cap {\sigma }_{w}\left(B\right)$. 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 ${\sigma }_{\tau }\left(A\right)\cup {\sigma }_{\tau }\left(B\right)$ to ${\sigma }_{\tau }\left({M}_{c}\right)$ is the punching of some open sets in ${\sigma }_{\tau }\left(A\right)\cap {\sigma }_{\tau }\left(B\right)$, where ${\sigma }_{\tau }\left(·\right)$ can be equal to the Browder spectrum or essential spectrum and the result is not true if ${\sigma }_{\tau }\left(·\right)$ is equal to the Kato spectrum, left (right) essential spectrum and left (right) spectrum.”

At the end, the authors present illustrative examples.

##### MSC:
 47A11 Local spectral properties 47A10 Spectrum 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