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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
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