WEIGHTED I-INFINITY NORMS FOR MATRICES

Let W = (omega(ij)) be a fixed n X n matrix of positive entries, and c onsider the W-weighted l(infinity) norm defined on C(n X n) by [GRAPHI CS] The main purpose of this note is to prove that for this norm, mult iplicativity, strong stability, and quadrativity are each equivalent t o the condition (W-1)2 less-than-or-equal-to W-1, where W-1 = (omega(i j)-1) is the Hadamard inverse of W. Among other things we also show th at if \\.\\W,infinity is k-bounded for some k greater-than-or-equal-to 2, then it is stable.