On the Berezin range of Toeplitz and weighted composition operators on weighted Bergman spaces

In this article, we completely characterize the Berezin range of Toeplitz operators with harmonic symbols acting on weighted Bergman spaces, illustrating the necessity of the harmonicity condition through examples. We then introduce a new class of weighted composition operators on these spaces, investigating their fundamental properties and determining their Berezin range and Berezin number. Finally, we study the convexity of the Berezin range of composition operators on weighted Bergman spaces and show that the origin lies in its closure of Berezin range but not in the range itself.