Generalized Berezin Number of Operators on Reproducing Kernel Hilbert Spaces

Let (KS,·,·) denote a reproducing kernel Hilbert space defined over a nonempty set S . For a given positive bounded linear operator A on KS , the A -Berezin number of an A -bounded linear operator T acting on KS is defined by berA(T)=μ| T\,vμ,vμA| , while the A -Berezin norm of T is defined by \|T\|_berA=μ,ν| Tvμ,vνA| . Here, vμ and vν are normalized reproducing kernels in KS , and ξ₁,ξ₂A= Aξ₁,ξ₂ for all ξ₁,ξ₂S . The main objective of this paper is to establish various inequalities involving the A -Berezin number berA(·) and the A -Berezin norm \|·\|_berA . In addition, we revisit and correct certain results recently published by Huban in Turkish J. Math., vol. 46, no. 1, 189–206 (2022). These corrections contribute to a more complete understanding of weighted reproducing kernel Hilbert spaces and highlight the symmetry inherent in the behavior of A -bounded linear operators.