The spectral unit ball \(\Omega_n\) is the set of all matrices
\(M\in \mathbb C^{n\times n}\) with spectral radius less than
\(1\). Let us call \(\pi\) the "projection" map which to a matrix
\(M\) associates \(\pi(M) \in \mathbb{C}^n\), the coefficients of
its characteristic polynomial (essentially), in fact the elementary
symmetric functions of its eigenvalues. Let \(\mathbb G_n:=
\pi(\Omega_n)\).
When investigating Pick-Nevanlinna problems for maps from the disk
to \(\Omega_n\), it is often useful to project the map to the
symmetrized polydisk (for instance to obtain continuity results for
the Lempert function, related to the two-point problem): if \(\psi
\in \mathcal O(\mathbb{D}, \Omega_n)\) and \(\psi (\alpha_j) =
M_j\), \(1\le j \le N\), then \(\pi \circ \psi \in \mathcal
O(\mathbb{D}, \mathbb G_n)\) and \(\pi \circ \psi (\alpha_j) =
\pi(M_j)\), \(1\le j \le N\). Given a map \(\varphi \in \mathcal
O(\mathbb{D}, \mathbb G_n)\), we are looking for necessary and
sufficient conditions for this map to "lift through given
matrices", i.e. find \(\psi\) as above so that \(\pi \circ \psi =
\varphi\). This is problematic when the matrices \(M_j\) are
derogatory (i.e. do not admit a cyclic vector). There are natural
necessary conditions, involving not only the values:
\(\varphi(\alpha_j)=\pi(M_j)\), of course, but also derivatives of
\(\varphi\) at the points \(\alpha_j\). Those conditions turn out
to be sufficient in small dimensions (up to \(4\)).
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