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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 $$aI \oplus bI \oplus\begin{pmatrix} aI & P \cr 0 & bI \cr\end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3)$$ forsome $a, b\in [0,1]$ and strictly positive operator $P$ with $\|P\| \le|\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary conditionfor 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 productof two positive contractions.

Factoring a quadratic operator as a product of two positive contractions