The hyperbolic cosine transform and its applications to composition operators

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 $C_{T,\rho }\text{\hspace{0.17em}⁣}:f\mapsto f\circ T$ on $L^2({\text{rbb}}^{\kappa },{\mu }_{\rho })$ with affine symbols $T=A+a$, where $\text{D}{\mu }_{\rho }(x)=\rho (x)\text{D}x$, $\rho (x)=\psi (\mathrm{\parallel }x\mathrm{\parallel }{)}^{-1}$, $\psi$ is a continuous positive real valued function and $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is the Euclidean norm on ${\text{rbb}}^{\kappa }$. The main result states that the map ${\text{rbb}}^{\kappa }\ni a\mapsto C_{I+a,\rho }$ 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.