Decomposition of a unitary scalar operator into a product of roots of the identity

Abstract

We prove that for all $m_1,m_2,m_3 \in \mathbb{N},~ \frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \leq 1$, every unitary scalar operator $\gamma I$ on a complex infinite-dimensional Hilbert space is a product $\gamma I = U_1 U_2 U_3$ where $U_i$ is a unitary operator such that $U_i^{m_i} = I$.