Berezin number and Berezin norm inequalities for operator matrices.

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 $ A=[A_{ij}] $ A = [ A i j ] is an $ n\times n $ n × n operator matrix with $ A_{ij}\in \mathbb {B}(\mathcal {H}) $ A i j ∈ B (H) for $ i,j=1,2,\dots, n $ i , j = 1 , 2 , ... , n , then $ \|A\|_{ber} \leq \left \|\left [\|A_{ij}\|_{ber}\right ]\right \| $ ‖ A ‖ b e r ≤ ‖ [ ‖ A i j ‖ b e r ] ‖ and $ \mathbf{ber}(A) \leq w([a_{ij}]), $ b e r (A) ≤ w ([ a i j ]) , where $ a_{ii}=\mathbf{ber}(A_{ii}), $ a i i = b e r (A i i) , $ a_{ij}=\big \||A_{ij}|+|A^*_{ji}|\big \|^{{1}/{2}}_{ber} \big \||A_{ji}|+|A^*_{ij}|\big \|^{{1}/{2}}_{ber} $ a i j = ‖ | A i j | + | A j i ∗ | ‖ b e r 1 / 2 ‖ | A j i | + | A i j ∗ | ‖ b e r 1 / 2 if i<j and $ a_{ij}=0 $ a i j = 0 if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space. [ABSTRACT FROM AUTHOR]