Some inequalities involving Hilbert-Schmidt numerical radius on $2\times 2$ operator matrices

Abstract

In this paper we present some inequalities related to the Hilbert-Schmidt numerical radius of $2\times2$ operator matrices. More precisely, we present a formula for the Hilbert-Schmidt numerical radius of an operator as follows:
\begin{align*}
w_{2}(T)=\sup_{\alpha^2+\beta^2=1}\|\alpha A+\beta B\|_{2},
\end{align*}
where $T=A+iB$ is the Cartesian decomposition of $T\in HS({\mathscr H})$.