Hilbert-Schmidt Numerical Radius of a Pair of Operators

We introduce a new norm on C-2 x C-2, where 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 x 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.

Automated summary

We introduce a new norm on C-2 x C-2, the Hilbert-Schmidt class, and study its basic properties and inequalities. As a result, new bounds for the Hilbert-Schmidt numerical radii of 2 x 2 operator matrices are derived. We also establish a connection with the classical Hilbert-Schmidt numerical radius of a single operator, and refine some existing bounds.