On Minimal Norms on $M_n$

Abstract

We show that for each minimal norm $N(\cdot )$ on the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, there exist norms $\| \cdot \|_1$ and $\| \cdot \|_2$ on $\mathbb{C}^n$ such that $N(A) = \max \{ \| Ax \|_2 : \| x \|_1 = 1, x\in \mathbb{C}^n \}$ for all $A \in \mathcal{M}_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.