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We provide one theorem of spectral equivalence of Koopman operators of anoriginal dynamical system and its reconstructed one through the delay-embeddingtechnique. The theorem is proved for measure-preserving maps (e.g. dynamics oncompact attractors) and provides a mathematical foundation of computingspectral properties of the Koopman operators by a combination of extendeddynamic mode decomposition and delay-embedding.

On the Spectral Equivalence of Koopman Operators through Delay Embedding