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

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