A-Numerical Radius of Semi-Hilbert Space Operators

Let $\mathbf{A=}\left(\!\! \begin{array}{cc} A & 0 \\ 0 & A \end{array}\!\! \right)$ be a $2\times 2$ diagonal operator matrix whose each diagonal entry is a positive bounded linear operator $A$ acting on a complex Hilbert space ${\mathcal{H}}$. Let $T,S$ and $R$ be bounded linear operators on ${\mathcal{H}}$ admitting $A$-adjoints, where $T$ and $R$ are $A$-positive. By considering an $\mathbf{A}$-positive $2 \times 2$ operator matrix $\left(\!\!\begin{array}{cc}T & S^{^{\sharp _{A}}} \\S & R \end{array}\!\!\right)$, we develop several upper bounds for the $A$-numerical radius of $S$. Applying these upper bounds we obtain new $A$-numerical radius bounds for the product and the sum of arbitrary operators which admit $A$-adjoints. Related other inequalities are also derived.