On the lattice property of the Koopman operator spectrum

TL;DR

This paper proves that the spectrum of the Koopman operator for discrete-time deterministic systems has a multiplicative lattice structure, while highlighting that stochastic systems may lack this property, clarifying key theoretical nuances.

Contribution

It establishes the lattice structure of the Koopman operator spectrum for discrete-time deterministic systems and distinguishes this from stochastic systems, advancing theoretical understanding.

Findings

Deterministic Koopman spectrum has a multiplicative lattice structure.

Stochastic Koopman spectrum does not necessarily have this structure.

Clarifies a fundamental property of Koopman operator theory.

Abstract

The Koopman operator has become a celebrated tool in modern dynamical systems theory for analyzing and interpreting both models and datasets. The linearity of the Koopman operator means that important characteristics about it, and in turn its associated nonlinear system, are captured by its eigenpairs and more generally its spectrum. Many studies point out that the spectrum of the Koopman operator has a multiplicative lattice structure by which eigenvalues and eigenfunctions can be multiplied to produce new eigenpairs. However, these observations fail to resolve whether the new eigenfunction remains in the domain of the Koopman operator. In this work, we prove that the spectrum of the Koopman operator associated to discrete-time dynamical systems has a multiplicative lattice structure. We further demonstrate that the Koopman operator associated to discrete-time stochastic process does not necessarily have such a structure, demonstrating an important nuance that lies at the heart of Koopman operator theory.