Berezin number and Berezin norm inequalities for operator matrices

Abstract

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if is an operator matrix with for , then and where if i<j and if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.