Green functions of the spectral ball and symmetrized polydisk

TL;DR

This paper studies the Green function of the spectral ball and symmetrized polydisk, revealing their relationships, differences, and estimates for invariant functions, especially in higher dimensions and for different types of matrices.

Contribution

It provides new insights into the behavior of Green functions on spectral balls and symmetrized polydisks, including explicit comparisons and estimates in various cases.

Findings

Green function is constant over isospectral varieties.

Equality holds for cyclic matrices, inequality is strict for derogatory matrices.

Invariant functions like Carathéodory distance differ in dimension ≥ 3.

Abstract

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality is always strict, and the difference between the two functions depends on the order of nilpotence of the strictly upper triangular blocks that appear in the Jordan decomposition of the pole. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carath\'eodory distance and the Green function, that show that they are distinct in dimension greater or equal to $3$.