Hilbert-Schmidt Numerical Radius of a Pair of Operators

Abstract

We introduce a new norm on \(\mathcal{C}_{2}\times \mathcal{C}_{2}\), where \(\mathcal{C}_{2}\) is the Hilbert-Schmidt class. We study basic properties of this norm and prove inequalities involving it. As an application of the present study, we deduce a chain of new bounds for the Hilbert-Schmidt numerical radii of \(2\times 2\) operator matrices. Connection with the classical Hilbert-Schmidt numerical radius of a single operator are also provided. Moreover, we refine some related existing bounds, too.