The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$

Abstract

We study a new domain in $\mathbb C^4$, namely the hexablock $\mathbb{H}$ that arises in connection with a special case of the $\mu$-synthesis problem in $M_2(\mathbb C)$. Previous attempts to study a few instances of the $\mu$-synthesis problem led to domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$. Throughout, the relations of $\mathbb{H}$ with these three domains are explored. Several characterizations of the hexablock are established. We determine the distinguished boundary $b \mathbb{H}$ of $\mathbb{H}$ and obtain a subgroup of the automorphism group of $\mathbb{H}$. The \textit{rational $\overline{\mathbb{H}}$-inner functions}, i.e., the rational functions from the unit disc $\mathbb{D}$ to $\mathbb{H}$ that map the boundary of $\mathbb{D}$ to $b\mathbb{H}$ are characterized. We obtain a Schwarz lemma for $\mathbb{H}$. Finally, it is shown that $\mathbb{G}_2$, $\mathbb{E}$, and $\mathbb{P}$ are analytic retracts of $\mathbb{H}$. By an application of the theory of the hexablock, we find alternative proofs to a few existing results on geometry and function theory of the pentablock.