On the Birman–Krein Theorem

Abstract

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.