Inequality between unitary orbits

Mitsuru Uchiyama; Michio Seto

doi:10.1016/j.crma.2013.04.024

Functional Analysis

Inequality between unitary orbits
[Inégalités entre orbites unitaires]

Présenté par : 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.

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 :

Mitsuru Uchiyama ¹ ; Michio Seto ¹

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

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     title = {Inequality between unitary orbits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {285--288},
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     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.024},
     language = {en},
}

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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/

Bibliographie
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