Parallelism and Rigorous Bounds for the A-Normalized Davis–Wielandt Berezin Number

In this paper, we introduce and study the A-normalized Davis–Wielandt Berezin number for bounded linear operators on a reproducing kernel Hilbert space. Suppose that the space is equipped with a positive semi-inner product induced by a non-zero positive semi-definite operator A. To avoid the geometric limitations of standard methods, we apply the A-normalized approach. We investigate the parallelism of A-bounded operators. Furthermore, we completely characterize A-Berezin normaloid operators. We establish the exact condition for equality in the triangle inequality related to the A-normalized Davis–Wielandt Berezin number. Finally, by using generalized Cauchy–Schwarz, Buzano, and Kittaneh–Moradi type inequalities, we deduce several new upper bounds for this quantity.