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Positivity of operator-matrices of Hua-type. (English) Zbl 1155.47019

Given ${A}_{1},\cdots ,{A}_{n}\in B\left(H\right)$ strict contractions (i.e., $\parallel {A}_{i}\parallel <1$), the author considers the operator matrix

${H}_{n}\left({A}_{1},\cdots ,{A}_{n}\right)={\left[{\left(I-{A}_{j}^{*}{A}_{i}\right)}^{-1}\right]}_{i,j=1}^{n}·$

L. K. Hua proved in [Acta Math. Sin. 5, 463–470 (1955; Zbl 0066.26601)] that ${H}_{2}\left({A}_{1},{A}_{2}\right)$ is positive-semidefinite. The author proved in [Linear Multilinear Algebra 8, 347–352 (1980; Zbl 0438.15019)] that the same is not necessarily true for ${H}_{3}\left({A}_{1},{A}_{2},{A}_{3}\right)$. In the paper under review, he shows a condition that guarantees positivity of ${H}_{n}$, and he shows that positivity of ${H}_{n}$ is preserved under the Möbius map (depending on the choice of a strict contraction $B$)

${{\Theta }}_{B}\left(Z\right)={\left(I-B{B}^{*}\right)}^{-1/2}\left(B-Z\right)\left(I-{B}^{*}Z\right){\left(I-{B}^{*}B\right)}^{1/2}·$

##### MSC:
 47A63 Operator inequalities 47B15 Hermitian and normal operators 15A45 Miscellaneous inequalities involving matrices
##### Keywords:
positivity; strict contraction; operator-matrix; Hua theorem