## 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

### Keywords:

matrix Reinhardt domain; holomorphically convex
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### References:

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