Discrete Spectrum Theorem
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We know that if two measure preserving automorphisms σ and τ are metrically isomorphic then the associated unitary operators U σ and U τ are unitarily equivalent. Let us say that σ and τ are spectrally isomorphic if U σ and U τ are unitarily equivalent. If σ and τ are spectrally isomorphic and σ is ergodic then τ is ergodic, because σ is ergodic if and only if 1 is a simple eigenvalue of U σ hence also of U τ , which in turn implies the ergodicity of τ. Similarly the mixing and weak mixing properties are invariant under spectral isomorphism. The question whether spectrally isomorphic measure preserving automorphisms are also metrically isomorphic has a negative answer. However in some special cases the answer is affirmative. One such situation is when U σ and U τ admit a complete set of eigenfunctions, σ and τ being ergodic and defined on a standard probability space.
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