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We introduce a new equivalence relation between unitary operators on separable Hilbert spaces and discuss a possibility to have in each equivalence class a measure-preserving transformation.
We show that for a unitary operator U on $L^2(X,μ)$, where X is a compact manifold of class $C^r$, $r ∈ ℕ ∪ {∞,ω}$, and μ is a finite Borel measure on X, there exists a function that realizes the maximal spectral type of U.
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