On the Birman-Krein Theorem

TL;DR

This paper proves that the only operator preserving singular spectral subspaces under all unitary conjugations is the identity, showing no broader generalization of the Birman-Krein Theorem is possible.

Contribution

It establishes a fundamental limitation on extending the Birman-Krein Theorem to preserve singular spectral subspaces under unitary transformations.

Findings

No nontrivial operators preserve singular spectral subspaces under all unitary conjugations.

The result characterizes the identity operator as uniquely preserving these subspaces.

It demonstrates the impossibility of a broader generalization of the Birman-Krein Theorem in this context.

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.