Koopman spectral analysis of skew-product dynamics on Hilbert $C^*$-modules

TL;DR

This paper develops a novel spectral analysis method for skew-product dynamical systems using a generalized eigenoperator decomposition on Hilbert $C^*$-modules, aiding in understanding system spectra and coherent structures.

Contribution

It introduces a new eigenoperator decomposition for Koopman operators on Hilbert $C^*$-modules, extending spectral analysis tools to skew-product systems.

Findings

Eigenoperator decomposition generalizes eigenvalue decomposition.

Reconstructs Koopman operator capturing continuous spectrum.

Numerical applications demonstrate method effectiveness.

Abstract

We introduce a linear operator on a Hilbert $C^*$-module for analyzing skew-product dynamical systems. The operator is defined by composition and multiplication. We show that it admits a decomposition in the Hilbert $C^*$-module, called eigenoperator decomposition, that generalizes the concept of the eigenvalue decomposition. This decomposition reconstructs the Koopman operator of the system in a manner that represents the continuous spectrum through eigenoperators. In addition, it is related to the notions of cocycle and Oseledets subspaces and it is useful for characterizing coherent structures under skew-product dynamics. We present numerical applications to simple systems on two-dimensional domains.