On the binary relation $ {⩽}_{u}$ on self-adjoint Hilbert space operators

Moslehian, M.S.; Nabavi Sales, S.M.S.; Najafi, H.

doi:10.1016/j.crma.2012.04.004

Functional Analysis

On the binary relation

⩽_{u}

on self-adjoint Hilbert space operators
[Relation binaire

⩽_{u}

sur un espace de Hilbert dʼopérateurs auto-adjoints]

Présenté par : the Editorial Board

Moslehian, M.S.¹ ; Nabavi Sales, S.M.S.¹ ; Najafi, H.¹

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

Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 407-410.

Résumé
Abstract

Soient $A, B \in B (H)$ des opérateurs auto-adjoints donnés, on dit que $A ⩽_{u} B$ si $A ⩽ U^{⁎} B U$ , où U est un opérateur unitaire. On montre que $A ⩽_{u} B$ si et seulement si $f (g {(A)}^{r}) ⩽_{u} f (g {(B)}^{r})$ pour toute fonction dʼopérateurs f, convexe et croissante, toute fonction dʼopérateurs g, monotone et tout nombre r positif. On donne des conditions nécessaires et suffisantes pour que $B ⩽ A ⩽ U^{⁎} B U$ implique $B = A = U^{⁎} B U$ . Enfin on montre que si $A^{n} ⩽ U^{⁎} A^{n} U$ pour tout $n \in N$ alors $A = U^{⁎} A U$ .

Given self-adjoint operators $A, B \in B (H)$ it is said $A ⩽_{u} B$ whenever $A ⩽ U^{⁎} B U$ for some unitary operator U. We show that $A ⩽_{u} B$ if and only if $f (g {(A)}^{r}) ⩽_{u} f (g {(B)}^{r})$ for any increasing operator convex function f, any operator monotone function g and any positive number r. We present some sufficient conditions under which if $B ⩽ A ⩽ U^{⁎} B U$ , then $B = A = U^{⁎} B U$ . Finally we prove that if $A^{n} ⩽ U^{⁎} A^{n} U$ for all $n \in N$ , then $A = U^{⁎} A U$ .

Reçu le : 2011-11-25
Accepté le : 2012-04-10
Publié le : 2012-04-01

DOI : 10.1016/j.crma.2012.04.004

Affiliations des auteurs :

Moslehian, M.S. ¹ ; Nabavi Sales, S.M.S. ¹ ; Najafi, H. ¹

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

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     author = {Moslehian, M.S. and Nabavi Sales, S.M.S. and Najafi, H.},
     title = {On the binary relation $ {\ensuremath{\leqslant}}_{u}$ on self-adjoint {Hilbert} space operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {407--410},
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Moslehian, M.S.; Nabavi Sales, S.M.S.; Najafi, H. On the binary relation $ {⩽}_{u}$ on self-adjoint Hilbert space operators. Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 407-410. doi : 10.1016/j.crma.2012.04.004. http://www.numdam.org/articles/10.1016/j.crma.2012.04.004/

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