Lifting maps from the symmetrized polydisk in small dimensions

TL;DR

This paper studies when maps into the symmetrized polydisk can be lifted to matrices with spectral properties, providing conditions for local and global lifts in small dimensions and highlighting limitations in higher dimensions.

Contribution

It establishes necessary and sufficient conditions for lifting maps from the symmetrized polydisk to matrices in small dimensions, extending previous results and identifying dimension-dependent limitations.

Findings

Necessary and sufficient conditions for local lifts involving derivatives.

A scheme for global lifts in dimensions up to 5.

Counter-example showing failure of the scheme in dimension 6 and above.

Abstract

The spectral unit ball $\Omega_n$ is the set of all $n\times n$ matrices with spectral radius less than $1$. Let $\pi(M) \in \mathbb C^n$ stand for the coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisk 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 polydisk (for instance to obtain continuity results for the Lempert function): if $\psi \in \mathcal O(\mathbb D, \Omega_n)$, then $\pi \circ \psi \in \mathcal O(\mathbb D, \mathbb G_n)$. 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 $\pi \circ \psi = \varphi$ and $\psi (\alpha_j) = M_j$, $1\le j \le N$. A natural necessary condition is $\varphi(\alpha_j)=\pi(M_j)$, $1\le j \le N$. When the matrices $M_j$ are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of $\varphi$ at the points $\alpha_j$. Those conditions are necessary and sufficient for a local lift. We give a scheme which shows that the necessary conditions are also sufficient for a global lift in small dimensions (up to $5$), and a counter-example to show that the scheme fails in dimension $6$ (and above).