On the Decomposition of Unitary Operators into a Product of Finitely Many Positive Operators

Abstract

We will show that in an infinite-dimensional separable Hilbert space $\mathcal{H}$, there exist constants $N\in \mathbb{N}$ and $c,d\in \mathbb{R}$ such that every unitary operator can be written as the product of at most $N$ positive invertible operators $\{a_k\}\subseteq B(\mathcal{H})$ with $\Vert a_k\Vert \le c$ and $\Vert a_k^{–1}\Vert \le d$ for all $k$. Some consequences of this result in the context of von Neumann algebras are discussed.