Annals of Functional Analysis

Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Qingying Bu, Chingjou Liao, and Ngai-Ching Wong

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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.

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

First available in Project Euclid: 7 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46G25: (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60]
Secondary: 17C65: Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] 46L05: General theory of $C^*$-algebras 47B33: Composition operators

Holomorphic functions homogeneous polynomial orthogonally additive and multiplicative zero product preserving matrix algebras


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.

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