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A note on Browder spectrum of operator matrices. (English) Zbl 1146.47004
For A(H), B(H) and C(K,H), define the operator M C acting on HK by M C =AC0B. In this paper the authors, by giving a counterexample, show that some recent results about Browder spectra of upper-triangular operator matrices are not always true and also, under some necessary and sufficient conditions, they determine the set σ b (M C ) over all C(K,H), where σ b (M C ) is the Browder spectrum of M C .
MSC:
47A10Spectrum and resolvent of linear operators
References:
[1]Zhang, Hai-Yan; Du, Hong-Ke: Browder spectra of upper-triangular operator matrices, J. math. Anal. appl. 323, 700-707 (2006) · Zbl 1109.47007 · doi:10.1016/j.jmaa.2005.10.073
[2]Zhang, H. -Y.; Du, H. -K.: Corrigendum to ”Browder spectra of upper-triangular operator matrices” [J. Math. anal. Appl. 323 (2006) 700 – 707], J. math. Anal. appl. 337, 751-752 (2007)
[3]Djordjevic, D. S.: Perturbations of spectra of operator matrices, J. operator theory 48, 467-486 (2002) · Zbl 1019.47003
[4]Cao, Xiao-Hong; Guo, Mao-Zheng; Meng, Bin: Drazin spectrum and Weyl’s theorem for operator matrices, J. math. Res. exposition 26, 413-422 (2006) · Zbl 1118.47004
[5]Taylor, A. E.; Lay, D. C.: Introduction to functional analysis, (1980) · Zbl 0501.46003
[6]Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. amer. Math. soc. 130, 1717-1723 (2002) · Zbl 0996.47015 · doi:10.1090/S0002-9939-01-06291-8