Basic Ergodic Theory pp 63-67 | Cite as

# Discrete Spectrum Theorem

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## Abstract

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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