Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices. (English) Zbl 1308.46055

Bu, Qingying; Liao, Chingjou; Wong, Ngai-Ching

Ann. Funct. Anal. 5, No. 2, 80-89 (2014).

Summary: 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 \[ H(x)= \sum_{n\geq 1} \lambda_n S^{-1} x^n S,\quad\forall x\in M_m, \] 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.

MSC:

46G25	(Spaces of) multilinear mappings, polynomials
17C65	Jordan structures on Banach spaces and algebras
46L05	General theory of \(C^*\)-algebras
47B33	Linear composition operators

Keywords:

holomorphic functions; homogeneous polynomials; orthogonally additive polynomial; multiplicative polynomial; zero product preserving; matrix algebras