Koopman Operators: Spectra, Function Spaces, and Representations

Igor Mezić ()
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Igor Mezić: University of California Santa Barbara, Mechanical Engineering and Mathematics

Chapter 104 in Operator Theory, 2026, pp 3143-3209 from Springer

Abstract: Abstract This chapter develops a unified spectral theory of Koopman (composition) operators for discrete- and continuous-time dynamical systems, emphasizing the interplay between operator theory, function spaces, and state-space geometry. We discuss the concept of representations of linear operators and formulate the representation eigenproblem. We then develop spectral theory on Banach and Hilbert spaces. For invertible measure-preserving systems, the Koopman operator is unitary on L 2 $$L^2$$ , and the spectral theorem yields a decomposition into pure point and continuous components. This leads to the Koopman Mode Decomposition, separating almost-periodic (structured) and mixing (random) dynamics, and clarifies the relation to invariant partitions and the Kronecker factor. Examples such as irrational rotations and the Arnold cat map illustrate pure point, Lebesgue, and mixing spectra. For uniformly hyperbolic (Anosov) systems, we discuss the emergence of continuous L 2 $$L^2$$ spectrum and introduce anisotropic Banach spaces on which revealing discrete resonances (Pollicott–Ruelle spectrum). In dissipative systems with Milnor attractors, we construct appropriate Hilbert and reproducing kernel spaces including modulated Fock spaces on which transient and asymptotic dynamics admit spectral expansions generated by principal Koopman eigenfunctions. The chapter concludes by connecting spectral type to the existence of finite-dimensional linear and nonlinear Koopman representations, highlighting conditions under which dynamical systems admit closed finite-dimensional models in observable space. Throughout, we emphasize the geometric meaning of eigenfunctions, their role in invariant decompositions, and the implications for data-driven modeling and operator-theoretic analysis of nonlinear dynamics.

Keywords: Koopman operator; Composition operator; Dynamical systems; Spectral theory of linear operators (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_124

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DOI: 10.1007/978-3-032-16356-1_124

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