The hyperbolic cosine transform and its applications to composition operators
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 $C_{T,ρ}\colon f \mapsto f \circ T$ on $L^2(\rbb^κ,μ_ρ)$ with affine symbols $T=A+a$, where $\D μ_ρ (x) = ρ(x) \D x$, $ρ(x)= ψ(\|x\|)^{-1}$, $ψ$ is a continuous positive real valued function and $\|\cdot\|$ is the Euclidean norm on $\rbb^κ$. The main result states that the map $\rbb^κ \ni a \mapsto C_{I+a,ρ}$ 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.