On A-numerical radius inequalities for $2\times 2$ operator matrices

Abstract

Let ($\mathcal{H},⟨.,.⟩)$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\Vert T{\Vert }_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H},⟨.,.{⟩}_A),$ where $⟨x,y{⟩}_A:=⟨Ax,y⟩$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\left[\begin{array}{ccc}A& 00& A\end{array}\right]$. Further, we prove some refinements of earlier $A$-numerical radius inequalities for operators.

16 pages