On the decomposition of unitary operators into a product of finitely many positive operators (Q1891938)

Summary: 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 \(\| a_ k\|\leq c\) and \(\| a_ k^{-1} \|\leq d\) for all \(k\). Some consequences of this result in the context of von Neumann algebras are discussed.