Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

Abstract

We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det (I-A_i^{\ast }{A}_j{)}^{-\alpha }{]}_{ij}$, where $A_i$ and $A_j$ are strict contractions and $\alpha \in \mathbb{R}$). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det (I-A^{\ast }B{)}^{-\alpha }$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.

This version incorporates all referee comments; we ran out of time to publish the work, but this version is close to the final revision that would have been sent to JFA (12 pages)

References in corpus (2)

• Conic geometric optimisation on the manifold of positive definite matrices
• The metric property of the quantum Jensen-Shannon divergence