Operator Sum Inequalities in the A-Normalized Berezin Framework

Building on recent approaches to generalized Berezin norms by by Albeladi, Feki, and Taha, this paper establishes several refined upper bounds for the A-normalized Berezin number and its associated seminorms for finite operator sums in reproducing kernel spaces, where A is a non-zero positive semidefinite operator inducing the semi-Hilbertian structure. By systematically leveraging the A-Cartesian decomposition alongside generalizations of the Dragomir, Buzano, and McCarthy inequalities, we deduce robust higher-power estimates that rectify and extend the existing literature. Furthermore, we derive an integral bound that refines the standard triangle inequality for the A-normalized Berezin operator radius, and establish a characterization of equality.