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Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces

Masahiro Ikeda, Isao Ishikawa, Corbinian Schlosser

TL;DR

This paper gives a general framework for Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces (RKBSs) and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time 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.

Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces

TL;DR

This paper gives a general framework for Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces (RKBSs) and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time 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.
Paper Structure (17 sections, 18 theorems, 119 equations)

This paper contains 17 sections, 18 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Koopman operators and reproducing kernel Banach spaces
  4. Review of Koopman and Perron-Frobenius operators
  5. Reproducing kernel Banach spaces
  6. Koopman and Perron-Frobenius operators on RKBS; discrete time systems
  7. Definitions of the Koopman and Perron-Frobenius operator
  8. Basic properties
  9. Examples
  10. Continuous time systems
  11. Koopman and Perron-Frobenius semigroup
  12. Koopman and Perron-Frobenius semigroups on RKBS
  13. Symmetry and sparsity patterns
  14. Conclusion
  15. Further notions with respect to the bilinear form
  16. ...and 2 more sections

Key Result

Lemma 3.9

Let $(\mathcal{B},\mathcal{B}',\langle\cdot,\cdot\rangle,k)$ be an RKBS on $X$ with kernel $k$ and $\phi:Y \rightarrow X$ be a bijective map. Then $(\mathcal{B}_\phi,\mathcal{B}_\phi',\langle\cdot,\cdot\rangle_\phi,k_\phi)$ is an RKBS on $Y$ with kernel for and with bilinear form and kernel where $\mathbb{K}$ denotes $\mathbb{R}$ or $\mathbb{C}$. Further the composition operator $T_\phi$ with

Theorems & Definitions (83)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Example 3.4
  • Definition 3.5: Reproducing kernel Banach spaceRKBSUnified
  • Definition 3.6: Kernels for RKBSRKBSUnified
  • Remark 3.7
  • Remark 3.8
  • Lemma 3.9: Pullback kernel
  • proof
  • ...and 73 more