The unique solution of the Karcher equation and the self-adjointness of the Karcher mean

Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution o...

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Bibliographic Details
Published in:Linear & multilinear algebra 2019-05, Vol.67 (5), p.976-986
Main Authors: Fujii, Jun Ichi, Seo, Yuki
Format: Article
Language:English
Subjects:
Banach spaces
geometric operator mean
Hilbert space
Karcher means
operator power mean
Primary: 47A64
Secondary: 47A63
Tsallis relative operator entropy
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Summary:Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2018.1440520