Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Abstract

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 $\{\lambda_n\}$ and an invertible $S$ in $M_m$ such that $$H(x) =\sum_{n\geq 1} \lambda_n S^{-1}x^nS, \quad\forall x \in M_m,$$ or $$H(x) =\sum_{n\geq 1} \lambda_n S^{-1}(x^t)^nS, \quad\forall x \in M_m.$$ 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.Comment: Ann. Funct. Anal. Volume 5, Number 2, 2014, to appear. This version enhances the published version with a note as "added in proofs