Consider \({\mathcal {H}}\) is a complex Hilbert space and A is a positive operator on \({\mathcal {H}}.\) The mapping \(\langle \cdot ,\cdot \rangle _A: {\mathcal {H}}\times {\mathcal {H}} \rightarrow {\mathbb {C}}\) , 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 \( \left\| \cdot \right\| _{A}\) . The A-Davis–Wielandt radius of an operator S on \({\mathcal {H}}\) is defined as \(d\omega _{A}\left( S\right) =\sup \left\{ \sqrt{\left| \left\langle Sz,z\right\rangle _{A}\right| ^{2}+\left\| Sz\right\| _{A}^{4}} :\left\| z\right\| _{A}=1\right\} .\) 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.
