Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B

Given a complex matrix equation AXA∗=B, where B∗=±B, we present explicit formulas for the maximal and minimal ranks of Hermitian (skew-Hermitian) solutions X to the equation as well as the maximal and minimal ranks of the real matrices X0 and X1 in a Hermitian (skew-Hermitian) solution X=X0+iX1. As applications, we give the maximal and minimal ranks of the real matrices C and D in a Hermitian (skew-Hermitian) g-inverse of a Hermitian (skew-Hermitian) matrix A+iB.