A generalization of the Davis–Wielandt radius for operators

Abstract

In this work, we define a generalization of the Davis–Wielandt radius of Hilbert space operators for an arbitrary norm and obtain some applications for Hilbert–Schmidt numerical radius inequalities. For an operator $T\in \mathscr {B} \left( \mathscr {H}\right)$, the Davis–Wielandt radius is defined as

$$\begin{aligned} dw_{N} \left( T\right) =\mathop {\sup }\limits _{\theta \in \mathbb {R}} \sqrt{ N^2\left( {{\mathfrak {R}} \left( {{\textrm{e}}^{i\theta } T} \right) } \right) + N^4\left( {{\mathfrak {R}} \left( {\mathrm{{e}}^{i\theta } |T|} \right) } \right) }. \end{aligned}$$

Using the generalization of the Davis–Wielandt radius, we present some properties of $dw_N{(\cdot )}$ and show several lower and upper bounds for $dw_N{(\cdot )}$. Moreover, we obtain some results for the Hilbert–Schmidt Davis–Wielandt radius inequalities.