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Weyl type theorem and hypercyclic property for bounded linear operators

  • School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119 ,China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.04.002
  • Received Date: 02 March 2020
  • Accepted Date: 03 April 2020
  • Rev Recd Date: 03 April 2020
  • Publish Date: 30 April 2020
    Abstract

  • Abstract

    Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. T∈B(H)satisfies Browder’s theorem if σ(T)\σw(T)π00(T) or σw(T)=σb(T). If σ(T)\σw(T)=π00(T), Weyl’s theorem holds for T, where σ(T), σw(T), σb(T) denote the spectrum set, Weyl spectrum, and Browder spectrum respectively, and π00(T)={λ∈iso σ(T):0<dim N (T－λI)<∞}. Using the newly defined spectrum, the sufficient and necessary conditions for operator functions satisfying Weyl type theorem were studied if T is a hypercyclic operator. In addition, the spectrum mapping theorem for some new spectrums was discussed.

    Abstract

    Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. T∈B(H)satisfies Browder’s theorem if σ(T)\σw(T)π00(T) or σw(T)=σb(T). If σ(T)\σw(T)=π00(T), Weyl’s theorem holds for T, where σ(T), σw(T), σb(T) denote the spectrum set, Weyl spectrum, and Browder spectrum respectively, and π00(T)={λ∈iso σ(T):0<dim N (T－λI)<∞}. Using the newly defined spectrum, the sufficient and necessary conditions for operator functions satisfying Weyl type theorem were studied if T is a hypercyclic operator. In addition, the spectrum mapping theorem for some new spectrums was discussed.
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    • References(12)
  • [1]
    WEYL H V. ber beschrnkte quadratische Formen, deren differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1): 373-392.
    [2]
    BERBERIAN S K. An extension of Weyl’s theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3): 273-279.
    [3]
    LI C, ZHU S, FENG Y, et al. Weyl’s theorem for functions of operators and approximation[J]. Integral Equations and Operator Theory, 2010, 67(4): 481-497.
    [4]
    CURTO R E, HAN Y M. Weyl’s theorem for algebraically paranormal operators[J]. Integral Equations and Operator Theory, 2003, 47(3):307-314.
    [5]
    AN I, HAN Y. Weyl’s theorem for algebraically quasi-class A operators[J]. Integral Equations and Operator Theory, 2008, 62(1): 1-10.
    [6]
    SHI W, CAO X. Weyl’s theorem for the square of operator and perturbations[J]. Communications in Contemporary Mathematics, 2015, 17(5): 36-46 .
    [7]
    COBURN L A. Weyl’s theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3): 285-288.
    [8]
    DUGGAL B P. The Weyl spectrum of p-hyponormal operators[J]. Integral Equations and Operator Theory, 1997, 29(2): 197-201.
    [9]
    CAO X. Analytically class operators and Weyl’s theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2): 795-803.
    [10]
    HARTE R, LEE W Y. Another note on Weyl’s theorem[J]. Trans Amer Math Soc, 1997, 349(5): 2115-2124.
    [11]
    HERRERO D A. Limits of hypercyclic and supercyclic operators[J]. Journal of Functional Analysis, 1991, 99(1): 179-190.
    [12]
    CAO X, GUO M, MENG B, et al. Weyl’s spectra and Weyl’s theorem[J]. Journal of Mathematical Analysis and Applications, 2003, 288(2): 758-767.)
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    [1]
    WEYL H V. ber beschrnkte quadratische Formen, deren differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1): 373-392.
    [2]
    BERBERIAN S K. An extension of Weyl’s theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3): 273-279.
    [3]
    LI C, ZHU S, FENG Y, et al. Weyl’s theorem for functions of operators and approximation[J]. Integral Equations and Operator Theory, 2010, 67(4): 481-497.
    [4]
    CURTO R E, HAN Y M. Weyl’s theorem for algebraically paranormal operators[J]. Integral Equations and Operator Theory, 2003, 47(3):307-314.
    [5]
    AN I, HAN Y. Weyl’s theorem for algebraically quasi-class A operators[J]. Integral Equations and Operator Theory, 2008, 62(1): 1-10.
    [6]
    SHI W, CAO X. Weyl’s theorem for the square of operator and perturbations[J]. Communications in Contemporary Mathematics, 2015, 17(5): 36-46 .
    [7]
    COBURN L A. Weyl’s theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3): 285-288.
    [8]
    DUGGAL B P. The Weyl spectrum of p-hyponormal operators[J]. Integral Equations and Operator Theory, 1997, 29(2): 197-201.
    [9]
    CAO X. Analytically class operators and Weyl’s theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2): 795-803.
    [10]
    HARTE R, LEE W Y. Another note on Weyl’s theorem[J]. Trans Amer Math Soc, 1997, 349(5): 2115-2124.
    [11]
    HERRERO D A. Limits of hypercyclic and supercyclic operators[J]. Journal of Functional Analysis, 1991, 99(1): 179-190.
    [12]
    CAO X, GUO M, MENG B, et al. Weyl’s spectra and Weyl’s theorem[J]. Journal of Mathematical Analysis and Applications, 2003, 288(2): 758-767.)
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