Factoring a quadratic operator as a product of two positive contractions

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 $$aI\oplus bI\oplus \left(\begin{array}{cc}aI& P\\ 0& bI\end{array}\right)\text{on}{H}_1\oplus H_2\oplus (H_3\oplus H_3)$$ for some $a,b\in [0,1]$ and strictly positive operator $P$ with $\Vert P\Vert \le |\sqrt{a}-\sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\left(\begin{array}{cc}{T}_1& T_3\\ 0& T_2\end{array}\right)$ on $H\oplus K$ that can be written as a product of two positive contractions.

9 pages