The magic functions and automorphisms of a domain (Q841657)

The authors introduce the notion of magic functions for a general domain \(\Omega\) in \({\mathbb C}^d\). Recall that the hereditary cone \(\text{Hered}(\Omega)\) of a domain \(\Omega\subset {\mathbb C}^d\) is defined as the set of all holomorphic functions \(h\in \text{Hol}(\Omega\times \overline{\Omega})\) such that \(h(T)\geq 0\) whenever \(T\) is a \(d\)-tuple of commuting operators on a Hilbert space, the joint spectrum of \(T\) is contained in \(\Omega\) and \(\| f(T)\|\leq \sup_{z\in\Omega}|f(z)|\), for every bounded function \(f\in \text{Hol}(\Omega)\). Recall also that, if \(h(x,y)=\sum c_{\alpha,\beta} y^\beta x^\alpha\) is the local expansion of \(h\) into a uniformly convergent power series, then, by the hereditary functional calculus, \(h(T)\) is defined (in multi-index notation) as the operator \(h(T)=\sum c_{\alpha,\beta} (T^*)^\beta T^\alpha\). A \textsl{magic function} on \(\Omega\) is by definition an analytic function \(f\in \text{Hol}(\Omega)\) such that the associated function \[ h_f(x,y)=1-\overline{f(\bar y)}f(x) \] lies on an extreme ray of the hereditary cone \(\text{Hered}(\Omega)\). The authors show that the family of magic functions of a given domain \(\Omega\) is an intrinsic complex-geometric object. In particular it is invariant under the group of holomorphic automorphisms Aut\((\Omega)\). They also obtain a formula for the Carathéodory pseudodistance on \(\Omega\) in terms of a set of generators of \(\text{Hered}(\Omega)\) of the form \(1-\overline{f(\bar y)}f(x)\), for \(f\in \text{Hol}(\Omega)\), and \(x,y\in \Omega\). Whenever such a set is compact, it contains a Carathéodory extremal function for \(x\) and \(y\). They explicitly determine the set of magic functions of the symmetrized bidisk \[ G=\{(z_1+z_2,z_1z_2)~|~ z_1,z_2\in \Delta\} . \] (Here, \(\Delta\) denotes the unit disc in \({\mathbb C}\)) which are compositions of a Möbius transformation with a function \[ \Phi_\omega: G\to {\mathbb C},\quad (s,p)\mapsto {{2\omega p-s}\over{2-\omega s}},\qquad (s,p)\in G,~ \omega\in S^1. \] Using the magic functions of \(G\), they determine the group of holomorphic automorphisms Aut\((G)\) (isomorphic to Aut\((\Delta)\)) [see also \textit{P. Pflug} and \textit{M. Jarnicki}, Arch. Math. 83, No.~3, 264--266 (2004; Zbl 1055.32020)] and they deduce a formula for the Carathédory pseudodistance of \(G\) [see also \textit{J. Agler} and \textit{N. J. Young}, J. Geom. Anal. 14, No.~3, 375--403 (2004; Zbl 1055.32010)].