A note on limits of unitarily equivalent operators

Abstract:

Let $\mathcal {U}(\mathcal {H})$ denote the set of all unitary operators on a separable complex Hilbert space $\mathcal {H}$. If T is a bounded linear operator on $\mathcal {H}$, let ${\pi _T}$ denote the mapping of $\mathcal {U}(\mathcal {H})$ onto $\mathcal {U}(T)$ given by conjugation. It is proved that if T is normal or isometric, then there exists a locally defined continuous cross-section for ${\pi _T}$ if and only if the spectrum of T is finite. Examples of nonnormal operators with local cross-sections are given.