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 $\| a_k \| ≤ c$ and $\|a^{–1}_k \| ≤ d$ for all $k$. Some consequences of this result in the context of von Neumann algebras are discussed.