Berezin number inequalities via positivity of 2×2 block matrices

Suppose B (H ()) is the set of all bounded linear operators acting on a reproducing kernel Hilbert space H () .Applying the positivity criteria of 22 block matrices, we develop several new upper bounds for the Berezin number of operators in B (H ()) involving Berezin norm, which are better than the earlier ones.Among other results, we obtain that if T,S B (H ()) and 0 < < 1 , then ber p (T S), for all p 1.By letting S = 0 , it follows that ber p (T ) 1 2 |T * | 2 p + |T | 2(1-)p ber , for all p 1.