Strongly Unitary Equivalence and Approximately Unitary Equivalence of Normal Compact Operators over Topological Spaces

TL;DR

This paper investigates conditions under which normal compact operators over topological spaces are unitarily equivalent or approximately unitarily equivalent, using obstruction theory and eigenvalue analysis.

Contribution

It provides a necessary and sufficient condition for strong unitary equivalence and a sufficient condition for approximate unitary equivalence when the base space is the circle.

Findings

Established a criterion for strong unitary equivalence of normal compact operators.

Provided a sufficient condition for approximate unitary equivalence on the circle.

Applied obstruction theory to analyze operator equivalences over topological spaces.

Abstract

Let $A$ and $B$ be compact operators over a topological space $X$ and suppose that these operators are normal and have same distinct eigenvalues at each point. By obstruction theory, we establish a necessary and sufficient condition for $A$ and $B$ to be strongly unitarily equivalent. When $X=S^1$, we also give a sufficient condition for $A$ and $B$ to be approximately unitarily equivalent with some assumption on their eigenvalues.