Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions
This solves a theoretical problem in functional analysis for researchers studying RKHS and composition operators, extending prior results to more general cases.
The paper determines which functions induce bounded composition operators on reproducing kernel Hilbert spaces (RKHS) with analytic positive definite functions, proving that only affine transforms do so in a broad class of RKHS, and shows these operators cannot be compact.
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.