Factoring a quadratic operator as a product of two positive contractions

TL;DR

This paper characterizes quadratic operators on complex Hilbert spaces that can be expressed as a product of two positive contractions, providing explicit structural conditions and necessary criteria for such factorizations.

Contribution

It offers a complete characterization of quadratic operators as products of positive contractions and introduces necessary conditions for certain block operators.

Findings

Quadratic operators can be factored into two positive contractions if they have a specific block structure.

Explicit bounds on the operator P ensure the factorization exists.

Necessary conditions are provided for block-structured operators to be products of 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.