On matrix Reinhardt domains. (English) Zbl 0672.32003

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.
Reviewer: X.Zhou


32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A99 Holomorphic functions of several complex variables
32E05 Holomorphically convex complex spaces, reduction theory
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