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 $$aI \oplus bI \oplus\begin{pmatrix} aI & P \cr 0 & bI \cr \end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3)$$ 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 $\begin{pmatrix} T_1 & T_3\ 0 & T_2\cr\end{pmatrix}$ on $H\oplus K$ that can be written as a product of two positive contractions.