A Family of Domains Associated with ${\mu}$ -Synthesis

Abstract

We introduce a family of domains—which we call the ${\mu_{1,\,n}}$ -quotients—associated with an aspect of ${\mu}$ -synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the ${\mu_{1,\,n}}$ -quotient and its associated unit “${\mu_E}$ -ball”. Here, ${\mu_E}$ is the structured singular value for the case ${E = \{[w]\oplus (z \mathbb{I}_{n - 1}) \in \mathbb{C}^{n \times n}: z, w \in \mathbb{C}\}, n = 2, 3, 4,\dots}$ Specifically: we show that, for such an E, the Nevanlinna–Pick interpolation problem with matricial data in a unit “${\mu_E}$ -ball”, and in general position in a precise sense, is equivalent to a Nevanlinna–Pick interpolation problem for the associated ${\mu_{1,\,n}}$ -quotient. Along the way, we present some characterizations for the ${\mu_{1,\,n}}$ -quotients.