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

Let $$\mathbb{A}$=\left( \begin{array}{cc} A & 0 \ 0 & A \ \end{array} \right)$bethe$2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that \begin{align*} w^2_{$\mathbb{A}$}\left(\begin{bmatrix} 0 & X \ Y & 0 \end{bmatrix}\right) &\leq \frac{1}{4}\max\Big{{\big|XX^{\sharp_A} + Y^{\sharp_A}Y\big|}{A}, {\big|X^{\sharp_A}X + YY^{\sharp_A}\big|}{A}\Big}

• \frac{1}{2}\max\big{w_{A}(XY), w_{A}(YX)\big}, \end{align*} where $w_{\mathbb{A}}(\cdot )$, ${\mathrm{\parallel }\cdot \mathrm{\parallel }}_A$ and $w_A(\cdot )$ are the $\mathbb{A}$-numerical radius, $A$-operator seminorm and $A$-numerical radius, respectively. We then apply the above inequality to find some upper bounds for the $\mathbb{A}$-numerical radius of certain $2\times 2$ operator matrices. In particular, we obtain some refinements of earlier $A$-numerical radius inequalities for semi-Hilbertian operators. An upper bound for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices of semi-Hilbertian space operators is also given.