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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 $$\| {\cdot }\| _1=\| {\cdot }\| _2$$.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
##### Keywords:
induced norm; algebra norm; full matrix algebra
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##### References:
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