Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric
We study positive definiteness of 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 $α\in\mathbb{R}$). 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…
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