Let \( T \) be a bounded linear operator acting on a reproducing kernel Hilbert space (RKHS) \( \mathcal {H} \) over a set \( X \) . The Berezin range of \( T \) is defined by \( \textrm{Ber}(T) := \{ \langle T \hat{k}_{x}, \hat{k}_{x} \rangle _{\mathcal {H}} : x \in X \}, \) where \( \hat{k}_{x} \) denotes the normalized reproducing kernel of \( \mathcal {H} \) at \( x \in X \) . In general, the Berezin range need not be convex. In this work, we investigate the convexity properties of the Berezin range for certain composition operators acting on classical functional Hilbert spaces, specifically the Fock space on \( \mathbb {C} \) and the Dirichlet space on the unit disc \( \mathbb {D} \) . We further establish an analogue of the elliptic range theorem for the unitarily equivalent Berezin range associated with operators on two-dimensional reproducing kernel Hilbert spaces. Moreover, we characterize the convexity of the unitarily equivalent Berezin range for a bounded operator \( T \) on a general RKHS. Finally, we provide a description of the invariant subspaces of the multiple shift operator in terms of the Berezin range of diagonal operators.
