On matrix Reinhardt domains
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.
scientific article; zbMATH DE number 4099662