Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball \(\Omega _n\) is the set of all \(n\times n\) matrices M with spectral radius less than 1. Let \(\pi (M) \in \mathbb {C}^n\) stand for the coefficients of the characteristic polynomial of a matrix M (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisc is \({{\mathbb {G}}}_n:=\pi (\Omega _n)\). When investigating Nevanlinna–Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisc (for instance to obtain continuity results for the Lempert function): if \(\Phi \in {\mathrm {Hol}}(\mathbb {D}, \Omega _n)\), then \(\pi \circ \Phi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\). Given a map \(\varphi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\), we are looking for necessary and sufficient conditions for this map to “lift through given matrices”, i.e. find \(\Phi \) as above so that \(\pi \circ \Phi = \varphi \) and \(\Phi (\alpha _j) = A_j\), \(1\le j \le N\). A natural necessary condition is \(\varphi (\alpha _j)=\pi (A_j)\), \(1\le j \le N\). When the matrices \(A_j\) are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of \(\varphi \) at the points \(\alpha _j\). We prove that those conditions are necessary and sufficient for a local lifting. We give a formula which performs the global lifting in small dimensions (\(n \le 5\)), and a counter-example to show that the formula fails in dimensions 6 and above.