×

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

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Abhyankar, S.S.: Local analytic geometry. New York: Academic Press 1964 · Zbl 0205.50401
[2] Grauert, H., Fritzsche, K.: Several complex variables. Berlin Heidelberg New York: Springer 1976 · Zbl 0381.32001
[3] Grauert, H., Remmert, R.: Plurisubharmonische Fuinktionen in komplexen R?umen. Math. Z.65, 175-194 (1956) · Zbl 0070.30403
[4] Henkin, G.M., Leiterer, S.: Theory of functions on complex manifolds. Basel Boston Stuttgart: Birkh?user 1984
[5] Krantz, S.G.: Function theory in several complex variables. New York: Wiley 1982 · Zbl 0471.32008
[6] Lelong, P., Gruman, L.: Entire functions of several complex variables. Berlin Heidelberg New York: Springer 1986 · Zbl 0583.32001
[7] Narashimhan, R.: Several complex variables. Chicago London: University of Chicago Press, 1971
[8] Sergeev, A.G.: On matrix Reinhardt domains. Preprint · Zbl 0778.32002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.