Advances in Operator Theory and Inequalities

abstract
Berezin numbers and their generalizations play a significant role in operator theory, particularly in the study of inequalities in Hilbert spaces. They also find applications in functional analysis, non-commutative geometry, and quantum physics. In this paper, we investigate several improvements and extensions of Berezin-type inequalities for bounded linear operators. We introduce generalized Euclidean Berezin numbers via convex functions and establish sharper bounds for classical inequalities. Furthermore, we extend weighted Davis–Wielandt inequalities and Furuta-type inequalities to the setting of semi-Hilbert spaces. These results not only enrich the theoretical framework of operator inequalities but also provide potential tools for further developments in mathematical physics and non-commutative analysis.