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 $E = \{[w]\oplus(z I_{n-1})\in \mathbb{C}^{n\times n}: z,w\in \mathbb{C}\}$, n = 2, 3, 4,... 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.