Consider $\mathcal{ H}$ is a complex Hilbert space and $A$ is a positive operator on $\mathcal{ H}.$ The mapping $\langle\cdot,\cdot\rangle_A: \mathcal{ H}\times \mathcal{ H} \to \mathbb { C}$, defined as $\left\langle y,z\right\rangle_{ A}=\left\langle Ay,z\right\rangle $ for all $y,z$ $\in $ ${ \mathcal{ H}}$, induces a seminorm $ \left\Vert \cdot\right\Vert_{ A}$. The $A$-Davis-Wielandt radius of an operator $S$ on $\mathcal{ H}$ is defined as $d\omega_{ A}\left( S\right) =\sup \left\{ \sqrt{ \left\vert \left\langle Sz,z\right\rangle_{ A}\right\vert ^{ 2}+\left\Vert Sz\right\Vert_{ A}^{ 4}} :\left\Vert z\right\Vert_{ A}=1\right\} \text{ .} $ We investigate some new bounds for $d\omega_{ A}\left( S\right)$ which refine the existing bounds. We also give some bounds for the $2\times 2$ off-diagonal block matrices.
Categories: math.FA