The hyperbolic cosine transform and its applications to composition operators

Abstract

In this paper we characterize hyperbolic cosine transforms of (positive) Borel measures ν in terms of exponential convexity (Bernstein's terminology). The case of compactly supported measures ν is also considered. All of this is then applied to (bounded) composition operators CT,ρ:f↦f∘T on L2(Rκ,μρ) with affine symbols T=A+a, where dμρ(x)=ρ(x)dx, ρ(x)=ψ(‖x‖)−1, ψ is a continuous positive real valued function and ‖⋅‖ is the Euclidean norm on Rκ. The main result states that the map Rκ∋a↦CI+a,ρ is continuous in the strong operator topology and has cosubnormal values if and only if ψ 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.