Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

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 Rd. We prove that only affine transforms can do so in a certain 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 intrinsic properties of the RKHSs, and we establish a connection between the behavior of composition operators and asymptotic properties of the greatest zeros of orthogonal polynomials on a weighted L2-space 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.