On a reverse of Ando--Hiai inequality

In this paper, we show a complement of Ando-Hiai inequality: Let A and B be positive invertible operators on a Hilbert space H and [alpha] [member of] [0,1]. If A [??] [alpha] [less than or equal to] B [less than or equal to] I, then [A.sup.r] [[??].sup.[alpha]] Br [less than or equal to] [[parallel...

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Bibliographic Details
Published inBanach journal of mathematical analysis Vol. 4; no. 1; pp. 87 - 91
Main Author Seo, Yuki
Format Journal Article
LanguageEnglish
Published Durham Springer 2010
Nature Publishing Group
Subjects
Hilbert space
Inequalities
Inequalities (Mathematics)
Mathematical analysis
Operator theory
Japan
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ISSN1735-8787
2662-2033
1735-8787
DOI10.15352/bjma/1272374672

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Summary:In this paper, we show a complement of Ando-Hiai inequality: Let A and B be positive invertible operators on a Hilbert space H and [alpha] [member of] [0,1]. If A [??] [alpha] [less than or equal to] B [less than or equal to] I, then [A.sup.r] [[??].sup.[alpha]] Br [less than or equal to] [[parallel](A [[??].sub.[alpha]] [B)].sup.-1] [parallel].sup.1-r] I for all 0 < r [less than or equal to] 1, where I is the identity operator and the symbol [parallel] x [parallel] stands for the operator norm. 2000 Mathematics Subject Classification. Primary 47A63; Secondary 47A30, 47A64. Key words and phrases. Ando-Hiai inequaqlity, positive operator, geometric mean.
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ISSN:1735-8787
2662-2033
1735-8787
DOI:10.15352/bjma/1272374672