Berezin number and Berezin norm inequalities for operator matrices

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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 i j ] is an n × n operator matrix with A i j ∈ B ( H ) for i , j = 1 , 2 , … , n , then ‖ A ‖ b e r ≤ ‖ [ ‖ A i j ‖ b e r ] ‖ and b e r ( A ) ≤ w ( [ a i j ] ) , where a i i = b e r ( A i i ) , 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 i j = 0 if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.