On the Berezin range and the Berezin radius of some operators

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over some non-empty set $\Omega$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle T\hat{k}_{\lambda},\hat{k}_{\lambda} \rangle_{\mathcal{H}} : \lambda \in \Omega\}$ and $\text{ber}(T)$ := $\sup\{|\gamma|: \gamma \in \text{Ber}(T)\}$, where $\hat{k}_{\lambda}$ is the normalized reproducing kernel for $\mathcal{H}(\Omega)$ 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 its Berezin set is discussed.