数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (6): 2259-2264.doi: 10.1016/S0252-9602(12)60175-1

• 论文 • 上一篇    下一篇

RIESZ IDEMPOTENT AND BROWDER´S THEOREM FOR ABSOLUTE-(p, r)-PARANORMAL OPERATORS

Salah Mecheri   

  1. Department of Mathematics, College of Science, Taibah University, P.O.Box 30002, Al Madinah Al Munawarah, Saudi Arabia
  • 收稿日期:2011-04-25 修回日期:2011-10-10 出版日期:2012-11-20 发布日期:2012-11-20
  • 基金资助:

    This work was supported by Taibah University Research Center Project (1433/803).

RIESZ IDEMPOTENT AND BROWDER´S THEOREM FOR ABSOLUTE-(p, r)-PARANORMAL OPERATORS

Salah Mecheri   

  1. Department of Mathematics, College of Science, Taibah University, P.O.Box 30002, Al Madinah Al Munawarah, Saudi Arabia
  • Received:2011-04-25 Revised:2011-10-10 Online:2012-11-20 Published:2012-11-20
  • Supported by:

    This work was supported by Taibah University Research Center Project (1433/803).

摘要:

An operator T is said to be paranormal if ||T2x|| ≥ ||Tx||2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈B(H) is called absolute-(p, r)-paranormal operator if |||T|p|T*|rx||r ≥ |||T*|rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0. The famous result of Browder, that self adjoint operators satisfy Browder´s theorem, is extended to several classes of operators. In this paper we show that for any absolute-(p, r)-paranormal operator T, T satisfies Browder´s theorem and a-Browder´s theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p, r)-paranormal operator T, then E is self-adjoint if and only if the null space of T μ, N(Tμ)  N(T* − μ).

关键词: absolute-(p, r)-paranormal operator, Browder´s theorem, Weyl´s spectrum

Abstract:

An operator T is said to be paranormal if ||T2x|| ≥ ||Tx||2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈B(H) is called absolute-(p, r)-paranormal operator if |||T|p|T*|rx||r ≥ |||T*|rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0. The famous result of Browder, that self adjoint operators satisfy Browder´s theorem, is extended to several classes of operators. In this paper we show that for any absolute-(p, r)-paranormal operator T, T satisfies Browder´s theorem and a-Browder´s theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p, r)-paranormal operator T, then E is self-adjoint if and only if the null space of T μ, N(Tμ)  N(T* − μ).

Key words: absolute-(p, r)-paranormal operator, Browder´s theorem, Weyl´s spectrum

中图分类号: 

  • 47B47