On matrix Reinhardt and circled domains.(English)Zbl 0778.32002

Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 573-586 (1993).
[For the entire collection see Zbl 0759.00008.]
The main purpose of this paper is to give a criterion for holomorphic convexity of matrix Reinhardt domains.
A domain $$D$$ in the space of $$n$$ matrix variables with $$m\times m$$ entries is said to be Reinhardt if for each point $$(Z_ 1,\ldots,Z_ n)\in D$$, the point $$(U_ 1Z_ 1V_ 1,\ldots,U_ nZ_ nV_ n)$$ also lies in $$D$$ for all unitary $$m\times m$$-matrices $$U_ j$$, $$V_ k$$. If $$(Z_ 1,\ldots,Z_ n)\in D\Rightarrow(UZ_ 1V,\ldots,UZ_ nV)\in D$$ for all unitary $$m\times m$$ matrices $$U$$, $$V$$, $$D$$ is said to be circled. Several examples of matrix Reinhardt and circled domains are given in the paper.
Given a matrix Reinhardt domain $$D$$, set $$\text{diag} D=\{(\Lambda_ 1,\ldots,\Lambda_ n)\in D:\Lambda_ i$$ is a diagonal matrix, $$1\leq i\leq n\}$$; diag $$D$$ is a (scalar) Reinhardt open set in the space $$\mathbb{C}^{mn}$$.
Theorem: Let $$D$$ be a complete matrix Reinhardt domain. Then $$D$$ is holomorphically convex if and only if diag $$D$$ is holomorphically convex.
Here completeness means that for each point $$(Z^ 0_ 1,\ldots,Z^ 0_ n)\in D$$, the matrix polydisk $\{(Z_ 1,\ldots,Z_ n):\| Z_ i\|\leq\| Z^ 0_ i\|,\qquad 1\leq i\leq n\}$ with respect to the spectral matrix norm $$\| Z\|=\max$$ {eigenvalues of $$\sqrt{Z*Z}\}$$ also lies in $$D$$.
Since, with the polar representation of a matrix in mind, it is natural to define the logarithmic image of $$D$$ as the logarithmic image of diag $$D$$, the author’s theorem can be considered as matrix analogue of the well-known criterion for holomorphic convexity of Reinhardt (scalar) domains. A similar result was proved independently by G. Khudajberganov [Mat. Vesn. 40, No. 3/4, 241-248 (1988; Zbl 0702.32001)].
Reviewer: J.Davidov (Sofia)

MSC:

 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 32D05 Domains of holomorphy

Citations:

Zbl 0759.00008; Zbl 0702.32001