Acta mathematica scientia, Series B >
RIESZ IDEMPOTENT AND BROWDER´S THEOREM FOR ABSOLUTE-(p, r)-PARANORMAL OPERATORS
Received date: 2011-04-25
Revised date: 2011-10-10
Online published: 2012-11-20
Supported by
This work was supported by Taibah University Research Center Project (1433/803).
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).
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 . DOI: 10.1016/S0252-9602(12)60175-1
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