Positivity of operator-matrices of Hua-type

Abstract

Let $A_j\,\,(j = 1, 2,\ldots , n)$ be strict contractions on a Hilbert space. We study an $n \times n$ operator-matrix: \[\textbf{H}_n(A_1,A_2,\ldots ,A_n) = [(I - A^*_j A_i)^{-1}]^n_{i,j=1}.\] For the case $n = 2$, Hua [Inequalities involving determinants, Acta Math. Sinica, 5 (1955), 463-470 (in Chinese)] proved positivity, i.e., positive semi-definiteness of $\textbf{H}_2(A_1,A_2)$. This is, however, not always true for $n = 3$. First we generalize a known condition which guarantees positivity of $\textbf{H}_n$. Our main result is that positivity of $\textbf{H}_n$ is preserved under the operator M\"obius map of the open unit disc $\mathcal D$ of strict contractions.