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

Given \(A_1,\dots,A_n\in B(H)\) strict contractions (i.e., \(\| A_i\| <1\)), the author considers the operator matrix \[ H_n(A_1,\dots,A_n)=[(I-A_j^*A_i)^{-1}]_{i,j=1}^n. \] L.K.Hua proved in [Acta Math.Sin.5, 463–470 (1955; Zbl 0066.26601)] that \(H_2(A_1,A_2)\) 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 \(H_3(A_1,A_2,A_3)\). 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 \(B\)) \[ \Theta_B(Z)=(I-BB^*)^{-1/2}(B-Z)(I-B^*Z)(I-B^*B)^{1/2}. \]

MSC:

47A63 Linear operator inequalities
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
15A45 Miscellaneous inequalities involving matrices
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