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

Let $\mathbb{B}(\mathcal{H})$ represent the $C^{\ast }$-algebra, which consists of all bounded linear operators on $\mathcal{H},$ and let $N(\mathrm{.})$ be a norm on $\mathbb{B}(\mathcal{H})$. We define a norm $w_{(N,e)}(\mathrm{.},\mathrm{.})$ on ${\mathbb{B}}^2(\mathcal{H})$ by $$w_{(N,e)}(B,C)=N\left(\mathrm{\Re }(e^{i\theta }({\lambda }_{1}B+{\lambda }_{2}C))\right),$$ for every $B,C\in \mathbb{B}(\mathcal{H})$ and ${\lambda }_1,{\lambda }_2\in \mathbb{C}\mathrm{.}$ We investigate basic properties of this norm and prove some bounds involving it. In particular, when $N(\mathrm{.})$ is the Hilbert-Schmidt norm, we prove some Hilbert-Schmidt Euclidean operator radius inequalities for a pair of bounded linear operators.