ar X iv : m at h / 04 07 35 7 v 1 [ m at h . FA ] 2 1 Ju l 2 00 4 Generalized Induced Norms ∗

Let ‖.‖ be a norm on the algebra Mn of all n × n matrices over C. An interesting problem in matrix theory is that ”are there two norms ‖.‖1 and ‖.‖2 on Cn such that ‖A‖ = max{‖Ax‖2 : ‖x‖1 = 1} for all A ∈ Mn. We will investigate this problem and its various aspects and will discuss under which conditions ‖.‖1 = ‖.‖2. 1 Preliminaries Throughout the paper Mn denotes the complex algebra of all n× n matrices A = [aij ] with entries in C together with the usual matrix operations. Denote by {e1, e2, · · · en} the standard basis for C, where ei has 1 as its ith entry and 0 elsewhere. We denote by Eij the n × n matrix with 1 in the (i, j) entry and 0 elsewhere. For 1 ≤ p ≤ ∞ the norm lp on C is defined as follows: