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

doi:10.1016/S0252-9602(12)60175-1

Abstract

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(T − μ) N(T* − μ).

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

CLC Number:

47B47

Salah Mecheri. RIESZ IDEMPOTENT AND BROWDER´S THEOREM FOR ABSOLUTE-(p, r)-PARANORMAL OPERATORS[J].Acta mathematica scientia,Series B, 2012, 32(6): 2259-2264.

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