Archiv der Mathematik ( IF 0.6 ) Pub Date : 2021-05-12 , DOI: 10.1007/s00013-021-01615-y Mohammad Sal Moslehian , Takashi Sano , Kota Sugawara
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.
中文翻译：
不定类型的算术几何平均不等式
本文讨论了不定类型的算术几何平均不等式。我们证明，对于满足\（I \ ge ^ JA \）和\（{\ mathrm {sp}}（A）\ subseteq [1，\ infty}，\）的J-自伴矩阵A
$$ \ begin {aligned} \ frac {I + A} {2} \ le ^ J \ sqrt {A} \ end {aligned} $$成立，对A则相反，它带有\（I \ ge ^ JA \）和\（{\ mathrm {sp}}（A）\ subseteq [0，1] \）。我们还证明，对于作用于任意维数的希尔伯特空间上的J正可逆算符A， B，不等式
$$ \ begin {aligned} \ frac {A + B} {2} \ ge ^ JA \ sharp ^ JB \ end {aligned} $$持有\（A \ sharp ^ JB：= J \ bigl（（JA）\ sharp（JB）\ bigr）\）。提供了一些涉及Pauli矩阵的示例以说明主要结果。
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