Essentially Commuting with a Unitary

TL;DR

This paper investigates the structure of unitaries that essentially commute with a given unitary operator with spectrum on the circle, showing the set is path-connected and characterizing related projections.

Contribution

It establishes the path-connectedness of unitaries essentially commuting with a spectrum-on-circle unitary and classifies the connected components of certain projections.

Findings

Set of unitaries essentially commuting with R is path-connected.

The set of projections commuting with R has connected components indexed by integers.

Provides a classification of projections based on their commutation properties.

Abstract

Let $R$ be a unitary operator whose spectrum is the circle. We show that the set of unitaries $U$ which essentially commute with $R$ (i.e., $[U,R]\equiv UR-RU$ is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with $R$ and obey a non-triviality condition, and prove it is bijective with $\mathbb{Z}$.