On a New Norm on the Space of Reproducing Kernel Hilbert Space Operators and Berezin Radius Inequalities

In this paper, we provide a new norm(a-Berezin norm) on the space of all bounded linear operators defined on a reproducing kernel Hilbert space, which generalizes the Berezin radius and the Berezin norm. We study the basic properties of the a-Berezin norm and develop various inequalities involving the a-Berezin norm. By using the inequalities we obtain various bounds for the Berezin radius of bounded linear operators, which improve on the earlier bounds. Further, we obtain a Berezin radius inequality for the sum of the product of operators, from which we derive new Berezin radius bounds.