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

Abstract

Let $\mathbb{B}(\mathcal{H})$ represent the $C^{\ast }$-algebra, which consists of all bounded linear operators on $\mathcal{H},$ and let $N(.)$ be a norm on $\mathbb{B}(\mathcal{H})$. We define a norm $w_{(N,e)}(.,.)$ on ${\mathbb{B}}^2(\mathcal{H})$ by $$w_{(N,e)}(B,C)=\underset{|{\lambda }_1{|}^2+{\lambda }_2{|}^2\le 1}{\sup }\underset{\theta \in \mathbb{R}}{\sup }N\left(\Re \left(e^{i\theta }({\lambda }_{1}B+{\lambda }_{2}C)\right)\right),$$ for every $B,C\in \mathbb{B}(\mathcal{H})$ 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.