Factoring a Quadratic Operator as a Product of Two Positive Contractions

Li, Chi-Kwong ; Tsai, Ming-Cheng

Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 354-362

Résumé

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.

Détail

DOI : 10.4153/CMB-2015-049-4

Mots-clés : 47A60, 47A68, 47A63, quadratic operator, positive contraction, spectral theorem