Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces

TL;DR

This paper extends the theoretical framework of Koopman and Perron-Frobenius operators from reproducing kernel Hilbert spaces to Banach spaces, introducing new properties and concepts applicable to dynamical systems analysis.

Contribution

It provides a general framework for these operators on RKBSs, expanding known properties from RKHSs and introducing new results like symmetry and sparsity for discrete and continuous systems.

Findings

Extended properties of Koopman and Perron-Frobenius operators to RKBSs

Introduced symmetry and sparsity concepts in the context of RKBSs

Applicable to both discrete and continuous time dynamical systems

Abstract

Koopman and Perron-Frobenius operators for dynamical systems have been getting popular in a number of fields in science these days. Properties of the Koopman operator essentially depend on the choice of function spaces where it acts. Particularly the case of reproducing kernel Hilbert spaces (RKHSs) draws more and more attention in data science. In this paper, we give a general framework for Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces (RKBSs). More precisely, we extend basic known properties of these operators from RKHSs to RKBSs and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time systems.