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

TL;DR

This paper characterizes bounded composition operators on RKHSs with analytic positive definite functions, showing only affine transformations induce such operators and exploring their compactness properties.

Contribution

It establishes that only affine transforms induce bounded composition operators on a broad class of RKHSs with analytic positive definite functions, extending previous results.

Findings

Only affine transforms induce bounded composition operators.

Bounded composition operators are not compact in this setting.

The method links composition operator behavior to zeros of orthogonal polynomials.

Abstract

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 $\mathbf{R}^d$. 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 $L^2$-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.