The hyperbolic cosine transform and its applications to composition operators

Abstract

In this paper we characterize hyperbolic cosine transforms of (positive) Borel measures $\nu$ in terms of exponential convexity (Bernstein's terminology). The case of compactly supported measures $\nu$ is also considered. All of this is then applied to (bounded) composition operators $C_{T,\rho}\colon f \mapsto f \circ T$ on $L^2(\rbb^\kappa,\mu_{\rho})$ with affine symbols $T=A+a$, where $\D \mu_{\rho} (x) = \rho(x) \D x$, $\rho(x)= \psi(\|x\|)^{-1}$, $\psi$ is a continuous positive real valued function and $\|\cdot\|$ is the Euclidean norm on $\rbb^{\kappa}$. The main result states that the map $\rbb^{\kappa} \ni a \mapsto C_{I+a,\rho}$ is continuous in the strong operator topology and has cosubnormal values if and only if $\psi$ is the hyperbolic cosine transform of a compactly supported Borel measure ($I$ is the identity transformation). The case of affine symbols $T$ that are not translations is also discussed.