  For a given bounded positive (semidefinite) linear operator $A$ on a complex
Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$, we
consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot\mid
\cdot\rangle_A \big)$ where ${\langle x\mid y\rangle}_A := \langle Ax\mid
y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an
$A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*}
\omega_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|\,; \,\,x\in
\mathcal{H}, \,{\langle x\mid x\rangle}_A= 1\Big\}. \end{align*} Our aim in
this paper is to derive several $\mathbb{A}$-numerical radius inequalities for
$2\times 2$ operator matrices whose entries are $A$-bounded operators, where
$\mathbb{A}=\text{diag}(A,A)$.
