Multiplicative preservers of C-numerical ranges and radii

Multiplicative preservers of C -numerical ranges and radii on certain groups and semigroups of complex n x n matrices are characterized. The general and special linear groups are considered, as well as the semigroups of matrices having ranks not exceeding k , with k fixed in advance. For a fixed C, it turns out that typically the multiplicative preservers of the C-numerical range (or radius) have the form A (bar right arrow) f(detA)UAU* or, for certain matrices C, the form A (bar right arrow) f (det A)U(A) over bar U*, for some unitary U and multiplicative map f from the group of nonzero complex numbers to the unit circle.