Multiplicative maps preserving the higher rank numerical ranges and radii

Let Mnbe the semigroup of n × n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mnincluding the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A ∈ Mn. Multiplicative maps φ{symbol} : S → Mnsatisfying rk(φ{symbol} (A)) = rk(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λk(A).