On Matrix-Valued Square Integrable Positive Definite Functions

TL;DR

This paper extends classical results on scalar positive definite functions to matrix-valued functions on unimodular groups, demonstrating their convolution structure and positivity properties using operator-theoretic methods.

Contribution

It generalizes Godement's results to matrix-valued functions, showing their convolution representation and positivity conditions in an operator-theoretic framework.

Findings

Matrix-valued positive definite functions can be expressed as convolutions of $L^2$ positive definite functions.

The integral of the trace of the product of two $L^2$ positive definite functions is non-negative.

Zero integral implies the convolution of the functions is zero.

Abstract

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 $\Phi$ and $\Psi$, $\int_G Trace(\Phi(g) \bar{\Psi(g)}^t) d g \geq 0$. In addition this integral equals zero if and only if $\Phi * \Psi=0$. Our proofs are operator-theoretic and independent of the group.