$A$-Davis-Wielandt Radius Bounds of Semi-Hilbertian Space Operators

Abstract

Consider $\mathcal{H}$ is a complex Hilbert space and $A$ is a positive operator on $\mathcal{H}.$ The mapping $⟨\cdot ,\cdot {⟩}_A:\mathcal{H}\times \mathcal{H}\to \mathbb{C}$, defined as ${⟨y,z⟩}_A=⟨Ay,z⟩$ for all $y,z$ $\in$ $\mathcal{H}$, induces a seminorm ${\Vert \cdot \Vert }_A$. The $A$-Davis-Wielandt radius of an operator $S$ on $\mathcal{H}$ is defined as $d{\omega }_A\left(S\right)=\sup \left\{\sqrt{{\left|{⟨Sz,z⟩}_A\right|}^2+{\Vert Sz\Vert }_A^4}:{\Vert z\Vert }_A=1\right\}\text{.}$ 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.