The unique solution of the Karcher equation and the self-adjointness of the Karcher mean
The unique solution of the Karcher equation and the self-adjointness of the Karcher mean
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Karcher方程的唯一解与Karcher均值的自伴性
DOI：
10.1080/03081087.2018.1440520
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发表时间：
2019
影响因子：
1.1
通讯作者：
Jun Ichi Fujii and Yuki Seo
中科院分区：
文献类型：
--
作者：
F. Hiai;Y. Seo and S. Wada;Masayuki Fujimoto and Yuki Seo;Yuki Seo;Yuki Seo;Jun Ichi Fujii;Jun Ichi Fujii and Takeaki Yamazaki;藤井淳一;Jun Ichi Fujii and Yuki Seo
Lawson和Lim利用Banach空间的隐函数定理证明了Hilbert空间上正可逆算子的Karcher方程有唯一解。本文在算子不等式的框架下，证明了Karcher方程唯一解的等价性和Karcher均值的自伴随性。为此，我们利用Tsallis负阶相对算子熵，改进了负阶算子幂手段的概念。
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.