Let $ \mathbb{ B}(\mathscr{ H})$ represent the $C^*$-algebra, which consists of all bounded linear operators on $\mathscr{ H},$ and let $N ( .) $ be a norm on $ \mathbb{ B}(\mathscr{ H})$. We define a norm $w_{ (N,e)} (. , . )$ on $ \mathbb{ B}^2(\mathscr{ H})$ by $$ w_{ (N,e)}(B,C)=\underset{ |\lambda_1|^2+\lambda_2|^2\leq1}\sup \underset{ \theta\in\mathbb{ R}}\sup N\left(\Re \left(e^{ i\theta}(\lambda_1B+\lambda_2C)\right)\right),$$ for every $B,C\in\mathbb{ B}(\mathscr{ H})$ and $\lambda_1,\lambda_2\in\mathbb{ C}.$ We investigate basic properties of this norm and prove some bounds involving it. In particular, when $N( .)$ is the Hilbert-Schmidt norm, we prove some Hilbert-Schmidt Euclidean operator radius inequalities for a pair of bounded linear operators.
Categories: math.FA