Sums of Murray–von Neumann equivalent operators

Jean-Christophe Bourin; Eun-Young Lee

doi:10.1016/j.crma.2013.09.019

Functional analysis

Sums of Murray–von Neumann equivalent operators

Presented by: Jean-Michel Bony

Jean-Christophe Bourin ¹; Eun-Young Lee ²

¹Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
²Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 761-764

Abstract (Original language)
Résumé

Let A, B be two Hilbert space positive operators such that $1 ⩾ ‖ B ‖ \neq 0$ and the positive part of $A - I$ satisfies $Tr {(A - I)}_{+} = \infty$ . Then $A = \sum_{n = 1}^{\infty} B_{n}$ , where $B_{n} \sim B$ for all n. ( $X \sim Y$ means $X = T T^{⁎}$ and $Y = T^{⁎} T$ .) This extends a 2009 result of Kaftal, Ng, and Zhang for sums of projections.

Si A est un opérateur positif tel que la partie positive de $A - I$ vérifie $Tr {(A - I)}_{+} = \infty$ , alors A est une somme de projections de rangs infinis. Ce résultat, obtenu en 2009 par Kalftal, Ng et Zhang, est étendu dans cette note aux sommes dʼopérateurs Murray–von Neumann équivalents à une contraction positive arbitraire.

Received: 2013-07-25
Accepted: 2013-09-25
Published online: 2013-10-01

DOI: 10.1016/j.crma.2013.09.019

Author's affiliations:

Jean-Christophe Bourin ¹ ; Eun-Young Lee ²

¹ Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
² Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

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     title = {Sums of {Murray{\textendash}von} {Neumann} equivalent operators},
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Jean-Christophe Bourin; Eun-Young Lee. Sums of Murray–von Neumann equivalent operators. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 761-764. doi: 10.1016/j.crma.2013.09.019

References
Cited by

[1] J.-C. Bourin Compressions and pinchings, J. Oper. Theory, Volume 50 (2003) no. 2, pp. 211-220

[2] J.-C. Bourin; E.-Y. Lee Unitary orbits of Hermitian operators with convex or concave functions, Bull. Lond. Math. Soc., Volume 44 (2012) no. 6, pp. 1085-1102

[3] J.-C. Bourin; E.-Y. Lee Decomposition and partial trace of positive matrices with Hermitian blocks, Int. J. Math., Volume 24 (2013) no. 1, p. 1350010 (13 p)

[4] K. Dykema; D. Freeman; K. Kornelson; D. Larson; M. Ordower; E. Weber Ellipsoidal tight frames and projection decompositions of operators, III, J. Math., Volume 48 (2004), pp. 477-489

[5] V. Kaftal; P.W. Ng; S. Zhang Strong sums of projections in von Neumann factors, J. Funct. Anal., Volume 257 (2009), pp. 2497-2529

[6] V. Kaftal; P.W. Ng; S. Zhang Projection decomposition in multiplier algebras, Math. Ann., Volume 352 (2012) no. 3, pp. 543-566

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