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