On the Birman–Krein Theorem - Bazao, Vanderléa R.; de Oliveira, César R.; Diaz, Pablo A. (Comptes Rendus. Mathématique)
It is shown that if $X$ is a unitary operator so that a singular subspace of $U$ is unitarily equivalent to a singular subspace of $UX$ (or $XU$), for each unitary operator $U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a singular spectral subspace in this context.