Geodesic
Generalized induced norms
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 127-133.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

Let $\Vert {\cdot }\Vert $ 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 $\Vert {\cdot }\Vert _1$ and $\Vert {\cdot }\Vert _2$ on ${\mathbb{C}}^n$ such that $\Vert A\Vert =\max \lbrace \Vert Ax\Vert _{2}\: \Vert x\Vert _{1}=1\rbrace $ for all $A\in {\mathcal M}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which $\Vert {\cdot }\Vert _1=\Vert {\cdot }\Vert _2$.
Classification : 15A60, 46B99, 47A30
Keywords: induced norm; generalized induced norm; algebra norm; the full matrix algebra; unitarily invariant; generalized induced congruent 
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Hejazian, S.; Mirzavaziri, M.; Moslehian, M. S. Generalized induced norms. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 127-133. http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a10/