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

Monire Hajmohamadi, Rahmatollah Lashkaripour

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 $T\in HS({\mathscr H})$.


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