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The relative operator entropy and the Karcher mean. (English) Zbl 1507.47041

Summary: The Karcher mean for positive invertible operators on a Hilbert space, which is the unique solution of the Karcher operator equation, was established by J. Lawson and Y.-D. Lim [Trans. Am. Math. Soc., Ser. B 1, 1–22 (2014; Zbl 1381.47010)] based on the geometric considerations for positive invertible operators. In this note, we extend both this mean and the equation to the non-invertible case. The key concept is the relative operator entropy. For operators whose ranges are closed, we show that the Karcher mean for the non-invertible case is the unique solution of the extended Karcher equation. As a byproduct, we can show the uniqueness of the Karcher solution for the invertible case based only on that of the power means.

MSC:

47A64 Operator means involving linear operators, shorted linear operators, etc.
47A63 Linear operator inequalities
47A58 Linear operator approximation theory
94A17 Measures of information, entropy

Citations:

Zbl 1381.47010
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References:

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