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

Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ 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 $w_{\mathbb{A}}(\cdot)$, ${\|\cdot\|}_{A}$ 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.