On matrix Reinhardt domains
type: Item
instance of: scholarly article
title: On matrix Reinhardt domains
zbMATH Open document ID: 0672.32003
DOI: 10.1007/BF01446876
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164668
review text: Let D be a domain in the space \({\mathbb{C}}^ n[m\times m]={\mathbb{C}}^{nm^ 2}\) of n matrix variables with \(m\times m\) entries, i.e. the points of D are given by n-tuples \(Z=(Z_ 1,...,Z_ n)\) where \(Z_ i\in {\mathbb{C}}[m\times m]\) are \(m\times m\) matrices with complex entries, \(i=1,...,n\). D is called a matrix Reinhardt domain, if \((Z_ 1,...,Z_ n)\in D\) implies that \((U_ 1Z_ 1V_ 1,...,U_ nZ_ nV_ n)\in D\) for arbitrary unitary matrices \(U_ j\), \(V_ k\) (1\(\leq j,k\leq n)\). We denote by diag D\(=\{(\Lambda_ 1,...,\Lambda_ n)\in D|\) \(\Lambda_ i\) are diagonal complex matrices, \(i=1,...,n\}.\) A. G. Sergeev posed the following conjecture: Let D be matrix Reinhardt domain. D is holomorphically convex if and only if \(Diag D\) is holomorphically convex (``only if'' part is trivial). In this paper, this conjecture is solved.
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32A99
Mathematics Subject Classification ID: 32E05
zbMATH DE Number: 4099662
zbMATH Keywords: matrix Reinhardt domain
zbMATH Keywords: holomorphically convex
author: Xiangyu Zhou
reviewed by: Xueyong Zhou
MaRDI profile type: Publication
cites work: Q5605672
cites work: Q4161937
cites work: Plurisubharmonische Funktionen in komplexen Räumen
cites work: Q3924489
cites work: Q3708270
cites work: Q4694620
description: scientific article; zbMATH DE number 4099662