Minimal circumscribed Hermitian ellipsoid of Hartogs domains and an application to an extremal problem (Q1043026)

In the present article a Hartogs domain \(W\) is defined to be a domain of the form \(W=\{(w,Y,Z)\in \mathbb C^N\times\mathbb C^{kl}\times\mathbb C^{mn}\,|\,\,\|w\|^2<\det(I-Y\overline{Y}^t)^{s_1}\cdot\det(I-Z\overline{Z}^t)^{s_2}\},\) with \(Y\) resp. \(Z\) in the Cartan domain of the first type \(\mathfrak{R}_I(k,l)\) resp. \(\mathfrak{R}_I(m,n)=\{Z\in \mathbb C^{(m,n)}\,|\,I-Z\overline{Z}^t>0\}\) and \(s_1\), \(s_2 >0\). A Hermitian ellipsoid \(E\) is a domain of the form \(E=\{z\in\mathbb C^p\,\,|\,\,\sum_{j,k=1}^p a_{jk}z_j\overline{z}_k<1\,\}\) with a positive definite Hermitian matrix \((a_{jk})\), i.e. the pre-image of the open unit ball \({\mathbf B}\subset\mathbb C^p\) under a complex linear automorphism of \(\mathbb C^p\). For any Hartogs domain \(W\) the authors give an explicit construction of the (uniquely determined) Hermitian ellipsoid with minimal volume which circumscribes \(W\). This ellipsoid is of the form \(E(a,b,c)=\{(w,Y,Z)\in\mathbb C^N\times\mathbb C^{kl}\times\mathbb C^{mn}\mid a\|w\|^2+b\|Y\|^2+c\|Z\|^2<1\}\). The construction of the ellipsoid leads to a solution of the so-called Carathéodory extremal problem for Hartogs domains, namely the existence of a holomorphic map \(f:W\rightarrow{\mathbf B}\) of \(W\) into the open unit ball \({\mathbf B}\subset\mathbb C^{N+kl+mn}\) with the property that \(\det |df(0)|=\sup\{|\det dg(0)|\mid g\in\text{ Hol }(W,{\mathbf B})\}\). For the construction of circumscribing ellipsoids of bounded symmetric domains and the Carathéodory extremal problem see also \textit{Y. Kubota} [Bull. Lond. Math. Soc. 15, 126--130 (1983; Zbl 0487.32018])], \textit{D. W. Ma} [J. Math. Soc. Japan 49, No. 4, 723--739 (1997; Zbl 0917.32017)] or \textit{A. Wang, X. Zhao} and \textit{Z. Y. Liu} [J. Korean Math. Soc. 44, No. 6, 1291--1312 (2007; Zbl 1144.32009)].