Reproducing kernel Hilbert space compactification of unitary evolution groups

Suddhasattwa Das, Dimitrios Giannakis, Joanna Slawinska

Research output: Contribution to journalArticlepeer-review


A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). The operator Wτ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, Wτ generates a unitary evolution group {etWτ}t∈R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of Wτ as τ→0+, as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.

Original languageEnglish (US)
Pages (from-to)75-136
Number of pages62
JournalApplied and Computational Harmonic Analysis
StatePublished - Sep 2021


  • Ergodic dynamical systems
  • Koopman operators
  • Perron-Frobenius operators
  • Reproducing kernel Hilbert spaces
  • Spectral theory

ASJC Scopus subject areas

  • Applied Mathematics


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