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Let $\mathbb{A}= \begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix} $ be a$2\times2$ diagonal operator matrix whose each diagonal entry is a boundedpositive (semidefinite) linear operator $A$ acting on a complex Hilbert space$\mathcal{H}$. In this paper, we derive several $\mathbb{A}$-numerical radiusinequalities for $2\times 2$ operator matrices whose entries are bounded withrespect to the seminorm induced by the positive operator $A$ on $\mathcal{H}$.Some applications of our inequalities are also given.

Further inequalities for the $\mathbb{A}$-numerical radius of certain $2 \times 2$ operator matrices