Composition operators, convexity of their Berezin range and related questions

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $\text{Ber}(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators $A$ on the reproducing kernel Hilbert space $\mathcal{H}(\Omega)$, namely we prove that $f(\mathrm{Ber}(\Phi(A)))=\mathrm{Ber}(\Phi(f(A)))$, where $\Phi:\mathcal{B}$ is a normalized positive linear map.