# Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

arXiv: https://arxiv.org/abs/1911.11992 · HTML: https://mlsanity.com/paper/1911.11992/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2019.

## Built on by (most cited first, 6 of 6)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [The kernel perspective on dynamic mode decomposition](https://mlsanity.com/paper/2106.00106/) (2021)
- [Finite-dimensional approximations of push-forwards on locally analytic functionals](https://mlsanity.com/paper/2404.10769/) (2024)
- [Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators](https://mlsanity.com/paper/2505.06806/) (2025)

## Abstract (arXiv)

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on Rd. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the Paley-Wiener space on the real line, studied in previous works, but also much more general RKHSs corresponding to analytic positive definite functions where existing methods do not work. Our method only relies on an intrinsic properties of the RKHSs, and we establish a connection between the behavior of composition operators and the asymptotic properties of the greatest zeros of orthogonal polynomials on a weighted L2-spaces on the real line. We also investigate the compactness of the composition operators and show that any bounded composition operators cannot be compact in our situation.