Bounds for the Davis-Wielandt radius of bounded linear operators

TL;DR

This paper establishes improved bounds for the Davis-Wielandt radius of bounded linear operators and operator matrices on complex Hilbert spaces, including exact values for specific matrix forms and new bounds for general matrices.

Contribution

It provides tighter bounds for the Davis-Wielandt radius, exact values for certain operator matrices, and extends the bounds to more general matrix forms, advancing the understanding of this operator measure.

Findings

Improved upper and lower bounds for the Davis-Wielandt radius.

Exact Davis-Wielandt radius for specific operator matrices.

New bounds for general operator matrices.

Abstract

We obtain upper and lower bounds for the Davis-Wielandt radius of bounded linear operators defined on a complex Hilbert space, which improve on the existing ones. We also obtain bounds for the Davis-Wielandt radius of operator matrices. We determine the exact value of the Davis-Wielandt radius of two special type of operator matrices $\left(\begin{array}{cc} I & B 0 & 0 \end{array}\right)$ and $\left(\begin{array}{cc} 0 & A B & 0 \end{array}\right)$, where $A,B\in \mathcal{B}(\mathcal{H})$, $I$ and $0$ are the identity operator and the zero operator on $\mathcal{H},$ respectively. Finally we obtain bounds for the Davis-Wielandt radius of operator matrices of the form $\left(\begin{array}{cc} A& B 0 & C \end{array}\right),$ where $A,B, C\in \mathcal{B}(\mathcal{H}).$