Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator

By Ryan Mohr, Igor Mezi'c. 2014

66 citations recorded by the source corpus.

This paper extends Yosida's mean ergodic theorem in order to compute projections onto non-unitary eigenspaces for spectral operators of scalar-type on locally convex linear topological spaces. For spectral operators with dominating point spectrum, the projections take the form of Laplace averages, which are a generalization of the Fourier averages used when the spectrum is unitary. Inverse iteration and Laplace averages project onto eigenspaces of spectral operators with minimal point spectrum. Two classes of dynamical systems --- attracting fixed points in $\mathbb{C}^{d}$ and attracting limit cycles in $\mathbb{R}^{2}$ --- and their respective spaces of observables are given for which the associated composition operator is spectral. It is shown that the natural spaces of observables are completions with an $\ell^{2}$ polynomial norm of a space of polynomials over a normed unital commutative ring. These spaces are generalizations of the Hardy spaces $H^{2}(\mathbb{D})$ and $H^{2}(\mathbb{D}^{d})$. Elements of the ring are observables defined on the attractor --- the fixed point or the limit cycle, in our examples. Furthermore, we are able to provide a (semi)global spectral theorem for the composition operators associated with a large class of dissipative nonlinear dynamical systems; any sufficiently smooth dynamical system topologically conjugate to either of the two cases above admits an observable space on which the associated Koopman operator is spectral. It is conjectured that this is generically true for systems where the basin of attraction can be properly "coordinatized".

Subfields: Control and Systems Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/77b17d697ebc64650f2767db40a4d4cac8facd75

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In Control and Systems Engineering

Igor Mezi'c

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Generalizing Koopman Theory to Allow for Inputs and Control (2016, unknown): “be defined by the Lebesque square-integrable functions, other appropriate spaces [34], or polynomial functions defined on a compact set.”
On Numerical Approximations of the Koopman Operator (2020, methodology): “Such results have previously been obtained in Hardy-type spaces [13], and here, we introduce a Gel’fand-formula-based technique that allows us to expand to general Banach spaces.”