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

Abstract

Let $\mathbb{A}=\left(\begin{array}{cc}A& 0\\ 0& A\end{array}\right)$ be a $2\times 2$ diagonal operator matrix whose each diagonal entry is a bounded positive (semidefinite) linear operator $A$ acting on a complex Hilbert space $\mathcal{H}$. In this paper, we derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator $A$ on $\mathcal{H}$. Some applications of our inequalities are also given.