Generalized Euclidean operator radius inequalities of a pair of bounded linear operators

Abstract

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 $$ w_{(N,e)}(B,C)=\underset{|λ_1|^2+λ_2|^2\leq1}\sup \underset{θ\in\mathbb{R}}\sup N\left(\Re \left(e^{iθ}(λ_1B+λ_2C)\right)\right),$$ for every $B,C\in\mathbb{B}(\mathscr{H})$ and $λ_1,λ_2\in\mathbb{C}.$ 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.