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Factoring a Quadratic Operator as a Product of Two Positive Contractions

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li  and
Ming-Cheng Tsai
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William & Mary, Williamsburg, VA 23187, USA e-mail: [email protected]
Ming-Cheng Tsai
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan e-mail: [email protected]
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Abstract

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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

$$aI\,\oplus \,bI\,\oplus \left( \begin{matrix} aI & P \\ 0 & bI \\ \end{matrix} \right)\,\text{on}\,{{H}_{1}}\,\oplus \,{{H}_{2}}\,\oplus \,\left( {{H}_{3\,}}\,\oplus \,{{H}_{3}} \right)$$

for some $a,\,b\,\in \,\left[ 0,\,1 \right]$ and strictly positive operator $P$ with $\left\| P \right\|\,\le \,\left| \sqrt{a}-\sqrt{b} \right|\sqrt{\left( 1-a \right)\left( 1-b \right)}$ . Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\left( \begin{matrix} {{T}_{1}} & {{T}_{3}} \\ 0 & {{T}_{2}} \\ \end{matrix} \right)$ on $H\,\oplus \,K$ that can be written as a product of two positive contractions.

Keywords

47A6047A6847A63quadratic operatorpositive contractionspectral theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 354 - 362
Copyright
Copyright © Canadian Mathematical Society 2016

References

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