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

TL;DR

This paper develops a method to compute spectral projections for scalar-type operators using Laplace averages, connecting these to Koopman operators in dynamical systems with attractors, and extends ergodic theorems to non-unitary spectra.

Contribution

It introduces a generalized approach to spectral projections via Laplace averages for scalar-type operators, applicable to Koopman operators in nonlinear dynamical systems.

Findings

Laplace averages generalize Fourier averages for non-unitary spectra.

Spectral projections can be computed for systems with attracting fixed points or limit cycles.

A (semi)global spectral theorem is established for a broad class of dissipative systems.

Abstract

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".