On minimal norms on \(M_n\)

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 b...

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Bibliographic Details
Published in:arXiv.org 2007-08
Main Authors: Mirzavaziri, Madjid, Mohammad Sal Moslehian
Format: Article
Language:English
Subjects:
Norms
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.0708.3358