For a selfadjoint involutive matrix J we equip ${\mathbb{C}}^n$ with the indefinite inner product $[x,y]=〈Jx,y〉$, where $〈\cdot ,\cdot 〉$ denotes the usual inner product on ${\mathbb{C}}^n$ and endow the matrix algebra ${\mathbb{M}}_n$ with a partially ordered relation $\stackrel{J}{\le }$. We define the notion of J-mean ${\sigma }_f(\cdot ,\cdot )$ of J-selfadjoint matrices with spectra in $(0,\infty )$ for any normal matrix monotone function $f:(0,\infty )\to (0,\infty )$. We establish several fundamental properties of J-means such as the power monotonicity and prove an indefinite version of the Ando–Hiai inequality.

中文翻译：

---

通过矩阵均值的不定矩阵不等式

对于自伴随对合矩阵J，我们装备${\mathbb{C}}^n$ 与不定内积 $[X,是]=〈JX,是〉$， 在哪里 $〈\cdot ,\cdot 〉$ 表示通常的内积 ${\mathbb{C}}^n$ 并赋予矩阵代数 ${\mathbb{米}}_n$ 具有偏序关系 $\stackrel{J}{\le }$. 我们定义J -mean的概念${\sigma }_F(\cdot ,\cdot )$的Ĵ在-selfadjoint矩阵使用光谱$(0,\infty )$ 对于任何正规矩阵单调函数 $F：(0,\infty )\to (0,\infty )$. 我们建立了J均值的几个基本性质，例如幂单调性，并证明了 Ando-Hiai 不等式的不确定版本。