ORTHOGONALLY ADDITIVE AND ORTHOGONALLY MULTIPLICATIVE HOLOMORPHIC FUNCTIONS OF MATRICES

Let H: Mm → Mm 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 H(x) = n≥1 λnS −1xnS, ∀x ∈Mm, or H(x) = n≥1 λnS −1(xt)nS, ∀x ∈Mm. Here, xt 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