Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Let $H:M_m\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\times m$ 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 $\{λ_n\}$ and an invertible $S$ in $M_m$ such that $H(x) =\sum_{n\geq 1} λ_n S^{-1}x^nS, \quad\forall x \in M_m,$ or $H(x) =\sum_{n\geq 1} λ_n S^{-1}(x^t)^nS, \quad\forall x \in M_m.$ 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.

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