\(\mathbb {A}\)-Berezin Number Inequalities for \(2\times 2\) Operator Matrices

Let \(\mathbb {A}\) $\mathbb{A}$ be the \(2\times 2\) $2\times 2$ diagonal operator matrix determined by a positive Hilbert space operator A. We give several upper bounds for the \(\mathbb {A}\) $\mathbb{A}$ -Berezin number of \(2\times 2\) $2\times 2$ block matrices on a reproducing kernel Hilbert space and prove inequalities for the A-Berezin number of Hilbert space operators. Our results in this paper generalize and refine earlier the A-Berezin number inequalities.