The group of automorphisms of the spectral ball (Q2884423)

Denote by \(M_n\) the set on \(n\times n\) complex matrices, and by \(\Omega_n\) the open subset of \(M_n\) consisting of matrices with spectral radius less than 1. The problem of describing the set \(\text{Aut}(\Omega_n)\) of all holomorphic automorphisms of \(\Omega_n\) was first addressed in [\textit{T. J. Ransford} and \textit{M. C. White}, ``Holomorphic self-maps of the spectral unit ball'', Bull. Lond. Math. Soc. 23, No. 3, 256--262 (1991; Zbl 0749.32018)], where the authors ask whether every element of \(\text{Aut}(\Omega_n)\) is a composition of maps of the following types: NEWLINENEWLINE{(i)}~ \(x\mapsto x^t\); NEWLINENEWLINE{(ii)}~ \(x\mapsto \gamma (x-\alpha I)(I-\overline{\alpha}x)^{-1}\), where \(|\gamma|=1\) and \(|\alpha|<1\); NEWLINENEWLINE{(iii)}~ \(x\mapsto u(x)^{-1}xu(x)\), where \(u\) is a holomorphic map from \(\Omega\) to the set \(M_n^{-1}\) of invertible matrices which satisfies \(u(qxq^{-1})=u(x)\) for all \(q\in M_n^{-1}\).NEWLINENEWLINENEWLINENEWLINEThis paper answers this question in the negative. Indeed, the author constructs two families of automorphisms of \(\Omega_2\) which cannot be written as compositions of maps of the forms (i), (ii) and (iii). These automorphisms are of the form \(x\mapsto u(x)xu(x)^{-1}\) with \(u\) holomorphic on \(\Omega_n\), but the maps \(u\) are not conjugation-invariant.