Inequality between unitary orbits

Mitsuru Uchiyama; Michio Seto

doi:10.1016/j.crma.2013.04.024

Functional Analysis

Inequality between unitary orbits

Presented by: Jean-Michel Bony

Mitsuru Uchiyama ¹; Michio Seto ¹

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

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

Abstract
Résumé

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.

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.

Received: 2013-03-18
Accepted: 2013-04-30
Published online: 2013-04-01

DOI: 10.1016/j.crma.2013.04.024

Author's affiliations:

Mitsuru Uchiyama ¹; Michio Seto ¹

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

@article{CRMATH_2013__351_7-8_285_0,
     author = {Mitsuru Uchiyama and Michio Seto},
     title = {Inequality between unitary orbits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {285--288},
     publisher = {Elsevier},
     volume = {351},
     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.024},
     language = {en},
}

TY  - JOUR
AU  - Mitsuru Uchiyama
AU  - Michio Seto
TI  - Inequality between unitary orbits
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 285
EP  - 288
VL  - 351
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2013.04.024
LA  - en
ID  - CRMATH_2013__351_7-8_285_0
ER  -

%0 Journal Article
%A Mitsuru Uchiyama
%A Michio Seto
%T Inequality between unitary orbits
%J Comptes Rendus. Mathématique
%D 2013
%P 285-288
%V 351
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2013.04.024
%G en
%F CRMATH_2013__351_7-8_285_0

Mitsuru Uchiyama; Michio Seto. Inequality between unitary orbits. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 285-288. doi : 10.1016/j.crma.2013.04.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.024/

References
Cited by

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

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

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

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

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

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

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

[8] M.S. Moslehian; S.M.S.N. Sales; H. Najafi 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] T. Okayasu; Y. Ueta 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] M.P. Olson The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Amer. Math. Soc., Volume 28 (1971), pp. 537-544

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

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

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

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

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

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

Cited by Sources:

Comments - Policy