# Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State Space Geometry

## TL;DR

This paper develops a spectral operator-theoretic framework for analyzing linear and nonlinear dynamical systems using Koopman operators, introducing new concepts like generalized eigenfunctions, open eigenfunctions, and spectral expansions in functional spaces.

## Contribution

It introduces a comprehensive spectral expansion approach for dynamical systems, extending Koopman analysis to nonlinear systems with attractors and defining new spectral and geometric concepts.

## Key findings

- Spectral expansions for systems with stable attractors and analytic observables.
- Characterization of invariant manifolds via Koopman eigenfunctions.
- Example of a measure-preserving system with continuous spectrum.

## Abstract

We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato Decomposition we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables, and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions - defined on subsets of state space - to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable and center-unstable manifolds in terms of joint zero level sets of families of Koopman operator eigenfunctions associated with the nonlinear system.After defining a new class of Hilbert spaces, that capture the on and off attractor properties of dissipative dynamics, and introduce the concept of Modulated Fock Spaces}, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. We provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.