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On minimal norms on $M_n$

Research paper by Madjid Mirzavaziri, Mohammad Sal Moslehian

Indexed on: 24 Aug '07Published on: 24 Aug '07Published in: Mathematics - Functional Analysis



Abstract

In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ 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 $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.