Some New Estimates for the Berezin Number of Hilbert Space Operators

In this paper, we have developed new estimates of some estimates involving the Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space H-Omega. The uniqueness or novelty of this article consists of new estimates of Berezin numbers for different types of operators. These estimates improve the upper bounds of the Berezin numbers obtained by other similar papers. We give several upper bounds for ber(T)(S+T), where T, S is an element of B(H-Omega) and r >= 1. We also present an estimation of ber(2r)( Sigma(d) (i=1) T-i) where T-i is an element of B(H-Omega), i =(SIC),(SIC) and r >= 1. Some of the obtained inequalities represent improvements to earlier ones. In this work, the ideas and methodologies presented may serve as a starting point for future investigation in this field.