JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (12): 59-66.doi: 10.6040/j.issn.1671-9352.0.2021.051

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Topological uniform descent and property (R) for bounded linear operators

ZHAO Xiao-peng1, DAI Lei1, CAO Xiao-hong2*   

  1. 1. School of Mathematics and Statistics, Weinan Normal University, Weinan 714099, Shaanxi, China;
    2. School of Mathematics and Statistics, Shaanxi Normal University, Xian 710119, Shaanxi, China
  • Published:2021-11-25

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Abstract: Let H be a complex separable infinite dimensional Hilbert space and B(H) be the algebra of all bounded linear operators on H, T∈B(H) is said to satisfy property (R) if σa(T)\σab(T)=π00(T), where σa(T) and σab(T)denote the approximate point spectrum and the Browder essential approximate point spectrum of T respectively, and π00(T)={λ∈iso σ(T):0N(T-λI)<∞}. By using the property of topological uniform descent, the necessary and sufficient conditions for which the property (R) holds for bounded linear operators are given. In addition, the new judgements for operator functions satisfying property (R) according to the property of topological uniform descent are discussed. Also, the property (R) for upper triangular operator matrices is explored.

Key words: topological uniform descent, property (R), spectrum

CLC Number: 

  • O177.2
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