On generalized Davis-Wielandt radius inequalities of semi-Hilbertian space operators

Let $A$ be a positive (semidefinite) operator on complex Hilbert space $\mathcal{H}$ and let $\mathbb{A}=\left(\begin{array}{cc} A & O O A \end{array}\right).$ We obtain upper lower bounds for the $A$-Davis-Wielandt radius of semi-Hilbertian operators, which generalize improve existing ones. also $\mathbb{A}$-Davis-Wielandt $2 \times 2$ matrices. Finally, we determine exact value two matrices $\left(\begin{array}{cc} I X 0 \end{array}\right)$ \end{array}\right)$, where $X $ is operator.