Banach Journal of Mathematical Analysis

Positivity of operator-matrices of Hua-type

Tsuyoshi Ando

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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.

Article information

Source
Banach J. Math. Anal. Volume 2, Number 2 (2008), 1-8.

Dates
First available in Project Euclid: 21 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1240336286

Digital Object Identifier
doi:10.15352/bjma/1240336286

Mathematical Reviews number (MathSciNet)
MR2391242

Zentralblatt MATH identifier
1155.47019

Subjects
Primary: 47B63
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 15A45: Miscellaneous inequalities involving matrices

Keywords
positivity strict contraction operator-matrix Hua theorem

Citation

Ando, Tsuyoshi. Positivity of operator-matrices of Hua-type. Banach J. Math. Anal. 2 (2008), no. 2, 1--8. doi:10.15352/bjma/1240336286. https://projecteuclid.org/euclid.bjma/1240336286


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