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.

15 pages; added Remark 3.6 and an additional reference; to appear in Integral Eqns. Operator Theory

Cited by in corpus (1)

• Geometric properties of domains related to $\mu$-synthesis