Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Let H : M_m → M_m be a holomorphic function of the algebra Mm of complex m×m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {λn} and an invertible S in Mm such that. Here, x^t is the transpose of the matrix x. In the latter case, we always have the first representation form when H also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.