Multiplicative maps preserving the higher rank numerical ranges and radii

Nung-Sing Sze
Sean Clark, William & Mary
Chi-Kwong Li, William & Mary


Let M(n) be the semigroup of n x n complex matrices under the usual multiplication, and let s be different subgroups or semigroups in M(n) including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Lambda(k)(A) is the rank-k numerical range and r(k)(A) is the rank-k numerical radius of A is an element of M(n). Multiplicative maps phi : S --> M(n) satisfying r(k)(phi(A)) = r(k)(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Lambda(k)(A). (C) 2009 Published by Elsevier Inc.