On matrix valued square integrable positive definite functions

Abstract

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two theorems of Godement on $L^2$ positive definite functions. We show that a matrix-valued continuous $L^2$ positive definite function can always be written as the convolution of a matrix-valued $L^2$ positive definite function with itself. We also prove that, given two $L^2$ matrix valued positive definite functions $\Phi$ and $\Psi$, $\int _G Tr(\Phi (g) \overline{\Psi (g)}^t) d g \ge 0$. In addition this integral equals zero if and only if $\Phi * \Psi =0$. Our proofs are operator-theoretic and independent of the group.