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The arithmetic-geometric mean inequality of indefinite type

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Abstract

In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying \(I \ge ^J A\) and \({\mathrm{sp}}(A) \subseteq [1, \infty ),\) the inequality

$$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$

holds, and the reverse does for A with \(I \ge ^J A\) and \({\mathrm{sp}}(A) \subseteq [0, 1]\). We also prove that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality

$$\begin{aligned} \frac{A + B}{2} \ge ^J A \sharp ^J B \end{aligned}$$

holds, where \(A \sharp ^J B:= J \bigl ( (JA) \sharp (JB) \bigr )\). Several examples involving Pauli matrices are provided to illustrate the main results.

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Acknowledgements

The authors would like to thank Professor Tsuyoshi Ando for fruitful comments to this article. They also thank the referee for several useful suggestions.

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Authors and Affiliations

  1. Department of Pure Mathematics Center of Excellence in Analysis of Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran

    Mohammad Sal Moslehian

  2. Faculty of Science, Yamagata University, Yamagata, 990-8560, Japan

    Takashi Sano

  3. Graduate School of Science and Engineering, Yamagata University, Yamagata, 990-8560, Japan

    Kota Sugawara

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  1. Mohammad Sal Moslehian
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  2. Takashi Sano
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  3. Kota Sugawara
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Correspondence to Mohammad Sal Moslehian.

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Moslehian, M.S., Sano, T. & Sugawara, K. The arithmetic-geometric mean inequality of indefinite type. Arch. Math. 117, 347–359 (2021). https://doi.org/10.1007/s00013-021-01615-y

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  • DOI: https://doi.org/10.1007/s00013-021-01615-y

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