Functional Analysis
Inequality between unitary orbits
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 285-288.

For bounded self-adjoint operators A and B we write $A⩽uB$ if there is a unitary U such that $A⩽U⁎BU$. In [7], Kosaki (1992) has shown that $A⩽B⇒exp(A)⩽uexp(B)$. In this note, we extend this; especially, we show that for a function $f(t)=∑i=1ncitaiebit$ with positive coefficients $ai$, $bi$ and $ci$, $0⩽A⩽uB⇒f(A)⩽uf(B)$. We then apply this to a positive linear map and get a similar inequality.

Pour deux opérateurs autoadjoints bornés A et B, nous écrirons $A⩽uB$ sʼil existe un opérateur unitaire U tel que $A⩽U⁎BU$. Kosaki (1992) a montré dans [7] que $A⩽B⇒exp(A)⩽uexp(B)$. Cette note étend ce résultat. En particulier nous montrons que pour les fonctions du type $f(t)=∑i=1ncitaiebit$ avec des coefficients $ai$, $bi$, $ci$ positifs, on a $0⩽A⩽uB⇒f(A)⩽uf(B)$. Ceci permet dʼobtenir des inégalités similaires pour les applications linéaires positives unitales.

Accepted:
Published online:
DOI: 10.1016/j.crma.2013.04.024

Mitsuru Uchiyama 1; Michio Seto 1

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

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