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# Factoring a Quadratic Operator as a Product of Two Positive Contractions

Published:2016-01-13
Printed: Jun 2016
• Chi-Kwong Li,
Department of Mathematics, College of William & Mary, Williamsburg, VA 23187, USA
• Ming-Cheng Tsai,
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
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## Abstract

Let $T$ be a quadratic operator on a complex Hilbert space $H$. We show that $T$ can be written as a product of two positive contractions if and only if $T$ is of the form \begin{equation*} aI \oplus bI \oplus \begin{pmatrix} aI & P \cr 0 & bI \cr \end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3) \end{equation*} for some $a, b\in [0,1]$ and strictly positive operator $P$ with $\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\big( \begin{smallmatrix} T_1 & T_3 \\ 0 & T_2\cr \end{smallmatrix} \big)$ on $H\oplus K$ that can be written as a product of two positive contractions.
 Keywords: quadratic operator, positive contraction, spectral theorem
 MSC Classifications: 47A60 - Functional calculus 47A68 - Factorization theory (including Wiener-Hopf and spectral factorizations) 47A63 - Operator inequalities

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