The isometries of certain maximum norms

Let A1, … , Am be nonsingular n × n matrices over F=R or C, and let 1 ⩽ pj ⩽ ∞ for j = 1, … , m. We characterize the linear isometries for the norm N on Fn defined byN(x)=max{lpj(Ajx):1⩽j⩽m},where lp(y) denotes the lp-norm of y∈Fn. As an application we study the conditions on matrices A1, … , Am for which N is absolute or a symmetric gauge function.