Linear maps transforming H-unitary matrices

Let H1 be an n×n invertible Hermitian matrix, and let U(H1) be the group of n×n H1-unitary matrices, i.e., matrices A satisfying A*H1A=H1. Suppose H2 is an m×m invertible Hermitian matrix. We show that a linear transformation φ:Mn→Mm satisfies φ(U(H1)) U(H2) if and only if there exist invertible matrices S Mm, U,V U(H2) such that S*H2S=[(Ia −Ib) H1] [(Ic −Id) (H1−1)t],and φ has the form A US[(Ia+b A) (Ic+d At)]S−1V,where a, b, c and d are nonnegative integers satisfying (a+b+c+d)n=m. Assume H1 has inertia (p,q) and H2 has inertia (r,s). Then there is a linear transformation mapping U(H1) into U(H2) if and only if there are nonnegative integers u and v such that (r,s)=u(p,q)+v(q,p). These results generalize those of Marcus, Cheung and Li.