Some conditions implying normality of operators

M.S. Moslehian; S.M.S. Nabavi Sales

doi:10.1016/j.crma.2011.01.018

Mathematical Analysis/Functional Analysis

Some conditions implying normality of operators

Presented by: the Editorial Board

M.S. Moslehian ¹; S.M.S. Nabavi Sales ^2,³

¹ Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
² Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
³ Tusi Mathematical Research Group (TMRG), Mashhad, Iran

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 251-254.

Abstract
Résumé

Let $T \in B (H)$ and $T = U | T |$ be its polar decomposition. We prove that (i) if T is log-hyponormal or p-hyponormal and $U^{n} = U^{⁎}$ for some n, then T is normal; (ii) if the spectrum of U is contained in some open semicircle, then T is normal if and only if so is its Aluthge transform $\tilde{T} = {| T |}^{\frac{1}{2}} U {| T |}^{\frac{1}{2}}$ .

Soit $T \in B (H)$ et $T = U | T |$ sa décomposition polaire. Nous montrons que : (i) si T est log-hyponormal ou p-hyponormal et $U^{n} = U^{⁎}$ pour un certain n, alors T est normal ; (ii) si le spectre de U est contenu dans un arc de cercle, alors T est normal si et seulement sʼil en est de même de son transformé de Aluthge $\tilde{T} = {| T |}^{1 / 2} U {| T |}^{1 / 2}$ .

Received: 2010-11-25
Accepted: 2011-01-18
Published online: 2011-02-02

DOI: 10.1016/j.crma.2011.01.018

Author's affiliations:

M.S. Moslehian ¹; S.M.S. Nabavi Sales ^{2, 3}

¹ Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
² Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
³ Tusi Mathematical Research Group (TMRG), Mashhad, Iran

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     title = {Some conditions implying normality of operators},
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M.S. Moslehian; S.M.S. Nabavi Sales. Some conditions implying normality of operators. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 251-254. doi : 10.1016/j.crma.2011.01.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.018/

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