## Annals of Functional Analysis

### 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 $$H(x) =\sum_{n\geq 1} \lambda_n S^{-1}x^nS, \quad\forall x \in M_m,%\eqno{(\ddag)}$$ 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.

#### Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 80-89.

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.afa/1396833505

Digital Object Identifier
doi:10.15352/afa/1396833505

Mathematical Reviews number (MathSciNet)
MR3192012

Zentralblatt MATH identifier
1308.46055

#### Citation

Bu, Qingying; Liao, Chingjou; Wong, Ngai-Ching. Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices. Ann. Funct. Anal. 5 (2014), no. 2, 80--89. doi:10.15352/afa/1396833505. https://projecteuclid.org/euclid.afa/1396833505

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