Inequality between unitary orbits

Uchiyama, Mitsuru; Seto, Michio

doi:10.1016/j.crma.2013.04.024

Functional Analysis

Inequality between unitary orbits
[Inégalités entre orbites unitaires]

Présenté par : Jean-Michel Bony

Uchiyama, Mitsuru ¹ ; Seto, Michio ¹

¹ Department of Mathematics, Shimane University, Matsue City, Shimane, Japan

Comptes Rendus. Mathématique, Tome 351 (2013) no. 7-8, pp. 285-288.

Résumé
Abstract

Pour deux opérateurs autoadjoints bornés A et B, nous écrirons $A \overset{u}{⩽} B$ sʼil existe un opérateur unitaire U tel que $A ⩽ U^{⁎} B U$ . Kosaki (1992) a montré dans [7] que $A ⩽ B \Rightarrow \exp (A) \overset{u}{⩽} \exp (B)$ . Cette note étend ce résultat. En particulier nous montrons que pour les fonctions du type $f (t) = \sum_{i = 1}^{n} c_{i} t^{a_{i}} e^{b_{i} t}$ avec des coefficients $a_{i}$ , $b_{i}$ , $c_{i}$ positifs, on a $0 ⩽ A \overset{u}{⩽} B \Rightarrow f (A) \overset{u}{⩽} f (B)$ . Ceci permet dʼobtenir des inégalités similaires pour les applications linéaires positives unitales.

For bounded self-adjoint operators A and B we write $A \overset{u}{⩽} B$ if there is a unitary U such that $A ⩽ U^{⁎} B U$ . In [7], Kosaki (1992) has shown that $A ⩽ B \Rightarrow \exp (A) \overset{u}{⩽} \exp (B)$ . In this note, we extend this; especially, we show that for a function $f (t) = \sum_{i = 1}^{n} c_{i} t^{a_{i}} e^{b_{i} t}$ with positive coefficients $a_{i}$ , $b_{i}$ and $c_{i}$ , $0 ⩽ A \overset{u}{⩽} B \Rightarrow f (A) \overset{u}{⩽} f (B)$ . We then apply this to a positive linear map and get a similar inequality.

Reçu le : 2013-03-18
Accepté le : 2013-04-30
Publié le : 2013-04-01

DOI : 10.1016/j.crma.2013.04.024

Affiliations des auteurs :

Uchiyama, Mitsuru ¹ ; Seto, Michio ¹

¹ Department of Mathematics, Shimane University, Matsue City, Shimane, Japan

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Uchiyama, Mitsuru; Seto, Michio. Inequality between unitary orbits. Comptes Rendus. Mathématique, Tome 351 (2013) no. 7-8, pp. 285-288. doi : 10.1016/j.crma.2013.04.024. http://www.numdam.org/articles/10.1016/j.crma.2013.04.024/

Bibliographie
Cité par

[1] Ando, T. On some operator inequalities, Math. Ann., Volume 279 (1987), pp. 157-159

[2] Bhatia, R. Matrix Analysis, Springer, 1996

[3] Bourin, J.C.; Lee, E.Y. Unitary orbits of Hermitian operators with convex or concave functions, Bull. Lond. Math. Soc., Volume 44 (2012), pp. 1085-1102

[4] Choi, M.D. A Schwarz inequality for positive linear maps on $C^{⁎}$ -algebras, Illinois J. Math., Volume 18 (1974), pp. 565-574

[5] Hansen, F.; Pedersen, G.K. Jensenʼs inequality for operators and Löwnerʼs theorem, Math. Ann., Volume 258 (1982), pp. 229-241

[6] Kadison, R.V. A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math., Volume 56 (1952), pp. 494-503

[7] Kosaki, H. On some trace inequalities, Proc. Centre Math. Appl. Austral. Nat. Univ., Volume 29 (1992), pp. 129-134

[8] Moslehian, M.S.; Sales, S.M.S.N.; Najafi, H. On the binary relation $⩽_{u}$ on self-adjoint Hilbert space operators, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012), pp. 407-410

[9] Okayasu, T.; Ueta, Y. A condition under which $B = A = U^{⁎} B U$ follows from $B ⩽ A ⩽ U^{⁎} B U$ , Proc. Amer. Math. Soc., Volume 135 (2007) no. 5, pp. 1399-1403

[10] Olson, M.P. The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Amer. Math. Soc., Volume 28 (1971), pp. 537-544

[11] Uchiyama, M. Commutativity of selfadjoint operators, Pacific J. Math., Volume 161 (1993), pp. 385-392

[12] Uchiyama, M. Perturbation and commutativity of unbounded selfadjoint operators, Bull. Fukuoka Univ. Ed. III, Volume 44 (1995), pp. 15-20

[13] Uchiyama, M. Strong monotonicity of operator functions, Integral Equations Operator Theory, Volume 37 (2000), pp. 95-105

[14] Uchiyama, M. Operator monotone functions which are defined implicitly and operator inequalities, J. Funct. Anal., Volume 175 (2000), pp. 330-347

[15] Uchiyama, M.; Hasumi, M. On some operator monotone functions, Integral Equations Operator Theory, Volume 42 (2002), pp. 243-251

[16] Uchiyama, M. A new majorization between functions, polynomials, and operator inequalities II, J. Math. Soc. Japan, Volume 60 (2008), pp. 291-310

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