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

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