Abstract
For a bounded linear operator A on a reproducing kernel Hilbert space H(Omega), with normalized reproducing kernel k circumflex expressionccent (lambda)=(k) over cap (lambda)/parallel to k(lambda)parallel to, the Berezin symbol, Berezin number and Berezin norm are defined respectively by A tilde (lambda)=(A) over tilde Ak circumflex (lambda),k circumflex, ber(A)=sup(lambda is an element of ohm)||A (k) over cap lambda||. A straightforward comparison between these characteristics yields the inequalities ber(A)<=parallel to A parallel to(ber) <=parallel to A parallel to. In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that ber(A)<=((parallel to)A parallel to(2)(ber) - inf lambda epsilon Omega parallel to (A - (A) over tilde (lambda))(k) over cap lambda parallel to(2))(1/2).