Let \( \mathbb {B}(\mathscr {H})\) represent the \(C^*\) -algebra, which consists of all bounded linear operators on \(\mathscr {H},\) and let \(N ( \cdot ) \) be a norm on \( \mathbb {B}(\mathscr {H})\) . We define a norm \(w_{(N,e)} (\cdot , \cdot )\) on \( \mathbb {B}^2(\mathscr {H})\) by \(\begin{aligned} w_{(N,e)}(B,C)=\underset{|\lambda _1|^2+|\lambda _2|^2\le 1}{\sup }\underset{\theta \in \mathbb {R}}{\sup }N\left( \Re \left( e^{i\theta }(\lambda _1B+\lambda _2C)\right) \right) \end{aligned}\) 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 \(N( \cdot )\) is the Hilbert-Schmidt norm, we prove some Hilbert-Schmidt Euclidean operator radius bounds for a pair of bounded linear operators.
