The unique solution of the Karcher equation and the self-adjointness of the Karcher mean |
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Authors: | Jun Ichi Fujii |
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Affiliation: | Department of Arts and Sciences (Information Science), Osaka Kyoiku University , Osaka, Japan. |
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Abstract: | 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. |
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Keywords: | Karcher means operator power mean Tsallis relative operator entropy geometric operator mean |
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