Convexity of the Berezin range of finite rank operators

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over a nonempty set $\Omega$, the Berezin range of $T$ is defined by \[ \mathrm{Ber}(T)=\left\{\langle T\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle_{\mathcal{H}} : \lambda \in \Omega \right\} \] and the Berezin radius is given by \[ \mathrm{ber}(T)=\sup\left\{ |\gamma| : \gamma \in \mathrm{Ber}(T) \right\}, \] where $\hat{k}_{\lambda}$ denotes the normalized reproducing kernel at $\lambda \in \Omega$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.