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On pseudo B-Weyl operators and generalized Drazin invertibility for operator matrices. (English) Zbl 1380.47010
Authors’ abstract: We introduce a new class which generalizes the class of B-Weyl operators. We say that \(T\in L(X)\) is pseudo B-Weyl if \(T=T_1\oplus T_2\), where \(T_1\) is a Weyl operator and \(T_2\) is a quasi-nilpotent operator. We show that the corresponding pseudo B-Weyl spectrum \(\sigma_{pBW}(T)\) satisfies the equality \(\sigma_{pBW}(T)\cup[S(T)\cap S(T^*)]=\sigma_{gD}(T)\), where \(\sigma_{gD}(T)\) is the generalized Drazin spectrum of \(T\in L(X)\) and \(S(T)\) (resp., \(S(T^*))\) is the set where \(T\) (resp., \(T^*\)) fails to have the SVEP. We also investigate the generalized Drazin invertibility of upper triangular operator matrices by giving sufficient conditions which assure that the generalized Drazin spectrum or the pseudo B-Weyl spectrum of an upper triangular operator matrices is the union of its diagonal entries’ spectra.

MSC:
47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
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References:
[1] Laursen KB, An introduction to local spectral theory (2000)
[2] Laursen KB, Czech. Math. J 39 pp 730– (1989)
[3] Mbekhta M, Studia Math 119 pp 129– (1996)
[4] DOI: 10.1007/BF02790238 · Zbl 0090.09003 · doi:10.1007/BF02790238
[5] DOI: 10.1007/BF02849344 · Zbl 0474.47008 · doi:10.1007/BF02849344
[6] DOI: 10.1090/S0002-9939-01-06291-8 · Zbl 0996.47015 · doi:10.1090/S0002-9939-01-06291-8
[7] DOI: 10.1007/BF01236475 · Zbl 0939.47010 · doi:10.1007/BF01236475
[8] Müller V, Proc. Workshop Geometry in Functional Analysis (2000)
[9] DOI: 10.1007/BF01351564 · Zbl 0177.17102 · doi:10.1007/BF01351564
[10] Amouch M, Math. Bohemica 136 pp 39– (2011)
[11] Zguitti H, Bull. Math. Anal. Appl 2 pp 27– (2010)
[12] Mbekhta M, C. R. Acad. Sci. Paris 303 série I 20 pp 979– (1986)
[13] Mbekhta M, J. Oper. Theory 24 pp 255– (1990)
[14] DOI: 10.1017/S0017089500031803 · Zbl 0897.47002 · doi:10.1017/S0017089500031803
[15] DOI: 10.1023/A:1013792207970 · Zbl 1079.47501 · doi:10.1023/A:1013792207970
[16] DOI: 10.4064/sm168-3-1 · Zbl 1071.47019 · doi:10.4064/sm168-3-1
[17] Apostol C, Rev. Roumaine Math. Pures Appl 19 pp 283– (1974)
[18] Taylor AE, Introduction to functional analysis (1980)
[19] DOI: 10.1090/surv/013/02 · doi:10.1090/surv/013/02
[20] DOI: 10.1155/IJMMS.2005.3497 · Zbl 1100.47028 · doi:10.1155/IJMMS.2005.3497
[21] Harte R, Invertibility and singularity for bounded linear operators (1988)
[22] DOI: 10.1002/1522-2616(200207)241:1<5::AID-MANA5>3.0.CO;2-F · Zbl 1030.46057 · doi:10.1002/1522-2616(200207)241:1<5::AID-MANA5>3.0.CO;2-F
[23] DOI: 10.1080/03081087.2012.698618 · Zbl 1275.47004 · doi:10.1080/03081087.2012.698618
[24] DOI: 10.1016/j.jmaa.2007.05.031 · Zbl 1148.47004 · doi:10.1016/j.jmaa.2007.05.031
[25] DOI: 10.1007/s00020-008-1648-8 · Zbl 1236.47004 · doi:10.1007/s00020-008-1648-8
[26] DOI: 10.1155/S0161171203012043 · Zbl 1060.47003 · doi:10.1155/S0161171203012043
[27] DOI: 10.1090/S0002-9939-99-04965-5 · Zbl 0944.47004 · doi:10.1090/S0002-9939-99-04965-5
[28] Houimdi M, Acta Math. Vietnam 25 pp 137– (2000)
[29] DOI: 10.1016/j.jmaa.2005.12.065 · Zbl 1105.47006 · doi:10.1016/j.jmaa.2005.12.065
[30] DOI: 10.1007/s00009-013-0275-z · Zbl 1304.47005 · doi:10.1007/s00009-013-0275-z
[31] DOI: 10.1016/j.laa.2008.06.002 · Zbl 1157.47004 · doi:10.1016/j.laa.2008.06.002
[32] DOI: 10.1007/s11766-014-3142-1 · Zbl 1313.47020 · doi:10.1007/s11766-014-3142-1
[33] DOI: 10.1016/j.jmaa.2015.02.037 · Zbl 1327.47001 · doi:10.1016/j.jmaa.2015.02.037
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