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Two-Variable Wasserstein Means of Positive Definite Operators

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Abstract

We investigate the two-variable Wasserstein mean of positive definite operators, as a unique positive solution of the nonlinear equation obtained from the gradient of the objective function of the least squares problem. A various properties of two-variable Wasserstein mean including the symmetry and the refinement of the self-duality are shown. Furthermore, interesting inequalities such as the Ando–Hiai inequality and bounds for the difference between the two-variable arithmetic and Wasserstein mean are provided. Finally, we explore the relationship between the tolerance relation and two-variable Wasserstein mean of positive definite Hermitian matrices.

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Acknowledgements

This research was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B 6001394).

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  1. Department of Mathematics, Chungbuk National University, Cheongju, 28644, Korea

    **mi Hwang & Sejong Kim

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  1. **mi Hwang
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  2. Sejong Kim
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Correspondence to Sejong Kim.

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Hwang, J., Kim, S. Two-Variable Wasserstein Means of Positive Definite Operators. Mediterr. J. Math. 19, 110 (2022). https://doi.org/10.1007/s00009-022-02011-8

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  • DOI: https://doi.org/10.1007/s00009-022-02011-8

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