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Generalized induced norms

S. Hejazian; M. Mirzavaziri; Mohammad Sal Moslehian — 2007

Czechoslovak Mathematical Journal

Let $∥ \cdot ∥$ be a norm on the algebra $ℳ_{n}$ of all $n \times n$ matrices over $ℂ$ . An interesting problem in matrix theory is that “Are there two norms ${∥ \cdot ∥}_{1}$ and ${∥ \cdot ∥}_{2}$ on $ℂ^{n}$ such that $∥ A ∥ = max {∥ A x ∥_{2} ∥ x ∥_{1} = 1}$ for all $A \in ℳ_{n}$ ?” We will investigate this problem and its various aspects and will discuss some conditions under which ${∥ \cdot ∥}_{1} = {∥ \cdot ∥}_{2}$ .