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Generalized induced norms. (English) Zbl 1174.15016
Summary: Let \(\| {\cdot }\| \) be a norm on the algebra \({\mathcal M}_n\) of all \(n\times n\) matrices over \({\mathbb C}\). An interesting problem in matrix theory is that “Are there two norms \(\| {\cdot }\| _1\) and \(\| {\cdot }\| _2\) on \({\mathbb C}^n\) such that \(\| A\| =\max \{\| Ax\| _{2}\: \| x\| _{1}=1\}\) for all \(A\in {\mathcal M}_n\)?” We investigate this problem and its various aspects and discuss some conditions under which \(\| {\cdot }\| _1=\| {\cdot }\| _2\).

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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