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.

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