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Zbl 1155.47019
Ando, Tsuyoshi
Positivity of operator-matrices of Hua-type.
(English)
[J] Banach J. Math. Anal. 2, No. 2, 1-8, electronic only (2008). ISSN 1735-8787/e

Given $A_1,\dots,A_n\in B(H)$ strict contractions (i.e., $\Vert A_i\Vert <1$), the author considers the operator matrix $$H_n(A_1,\dots,A_n)=[(I-A_j^*A_i)^{-1}]_{i,j=1}^n.$$ {\it L.\,K.\thinspace 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 $B$) $$\Theta_B(Z)=(I-BB^*)^{-1/2}(B-Z)(I-B^*Z)(I-B^*B)^{1/2}.$$
[Mart\'in Argerami (Regina)]
MSC 2000:
*47A63 Operator inequalities, etc.
47B15 Hermitian and normal operators
15A45 Miscellaneous inequalities involving matrices

Keywords: positivity; strict contraction; operator-matrix; Hua theorem

Citations: Zbl 0066.26601; Zbl 0438.15019

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