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

Abstract

We study positive definite functions on noncommuting strict contractions. In particular, we study functions that induce positive definite Hua-Bellman matrices (i.e., matrices of the form [det(I − A * i A j ) −α ] ij where A i and A j are strict contractions and α ∈ C). We start by revisiting a 1959 work of Bellman (R. Bellman Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies Hua-Bellman matrices and claims a strengthening of Hua's representation theoretic results on their positive definiteness (L.-K. Hua, Inequalities involving determinants; Acta Mathematica Sinica, 5(1955), pp. 463-470). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date. We "fix" this error and provide conditions under which det(I − A * B) −α is a positive definite function; our conditions correct Bellman's claim and subsume both Bellman's and Hua's prior results. Subsequently, we build on our result and introduce a new hyperbolic-like geometry on noncommuting contractions, and remark on its potential applications.