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

Given A 1 ,,A n B(H) strict contractions (i.e., A i <1), the author considers the operator matrix

H n (A 1 ,,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)

Θ B (Z)=(I-BB * ) -1/2 (B-Z)(I-B * Z)(I-B * B) 1/2 ·

47A63Operator inequalities
47B15Hermitian and normal operators
15A45Miscellaneous inequalities involving matrices