关于矩阵值平方可积〔的〕正定函数
On Matrix-Valued Square Integrable Positive Definite Functions
摘要：
本文研究了幺模群上的矩阵值正定函数。将Godement关于平方可积〔的〕正定函数的两个重要结果推广到矩阵值平方可积〔的〕正定函数.我们证明了一个矩阵值连续$L^2$正定函数总是可以写为一个$L^2$正定函数与自身的卷积。我们还证明了，给定两个$L^2$矩阵值正定函数$Φ$和$Ψ$，$\int_G Trace(Φ(g) \bar{Ψ(g)}^t) d g \geq 0$ .此外，当且仅当$Φ* Ψ=0$时，该积分等于零。我们的证明是算子理论和独立的组。
英文摘要：
In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite functions. We show that a matrix-valued continuous $L^2$ positive definite function can always be written as a convolution of a $L^2$ positive definite function with itself. We also prove that, given two $L^2$ matrix valued positive definite functions $Φ$ and $Ψ$, $\int_G Trace(Φ(g) \bar{Ψ(g)}^t) d g \geq 0$. In addition this integral equals zero if and only if $Φ* Ψ=0$. Our proofs are operator-theoretic and independent of the group.
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