On $\mathbb{A}$-numerical radius inequalities of operators and operator matrices

Let $\mathcal{H}$ be a complex Hilbert space and let $A$ be a positive operator on $\mathcal{H}$. We obtain new bounds for the $A$-numerical radius of operators in semi-Hilbertican space $\mathcal{B}_A(\mathcal{H})$. Further, we develop inequalities for the $\mathbb{A}$-numerical radius of $n\times n$ operator matrices of the form $(T_{ij})_{n\times n}$, where $T_{ij} \in \mathcal{B}_A(\mathcal{H})$ and $\mathbb{A}=\mbox{diag}(A,A,\ldots,A)$ is an $n\times n$ operator diagonal matrix. Finally, we estimate bounds for the $B$-operator seminorm and $B$-numerical radius of $2\times 2$ operator matrices, where $B=\mbox{diag}(A,A)$. The inequalities and bounds obtained here generalize and improve on the existing ones, respectively.