Convexity of the Berezin range of operators on $\mathcal{H}_\gamma (\mathbb{D})$

In this paper, we characterize the convexity of the Berezin range for finite-rank operators acting on the weighted Hardy space $\mathcal{H}_\gamma (\mathbb{D})$ over the unit disc $\mathbb{D}$. We provide a complete classification in terms of convexity for concrete operators. Additionally, we address dynamical properties of finite-rank operators on Hardy and Bergman spaces. Several illustrative examples are discussed to support our theoretical findings. Additionally, geometrical interpretations have also been employed.