Comptes Rendus
Functional Analysis
On the binary relation u on self-adjoint Hilbert space operators
Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 407-410.

Given self-adjoint operators A,BB(H) it is said AuB whenever AUBU for some unitary operator U. We show that AuB if and only if f(g(A)r)uf(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 BAUBU, then B=A=UBU. Finally we prove that if AnUAnU for all nN, then A=UAU.

Soient A,BB(H) des opérateurs auto-adjoints donnés, on dit que AuB si AUBU, où U est un opérateur unitaire. On montre que AuB si et seulement si f(g(A)r)uf(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 BAUBU implique B=A=UBU. Enfin on montre que si AnUAnU pour tout nN alors A=UAU.

Published online:
DOI: 10.1016/j.crma.2012.04.004

M.S. Moslehian 1; S.M.S. Nabavi Sales 1; H. Najafi 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
     author = {M.S. Moslehian and S.M.S. Nabavi Sales and H. Najafi},
     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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     year = {2012},
     doi = {10.1016/j.crma.2012.04.004},
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M.S. Moslehian; S.M.S. Nabavi Sales; H. Najafi. On the binary relation $ {⩽}_{u}$ on self-adjoint Hilbert space operators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 407-410. doi : 10.1016/j.crma.2012.04.004.

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