A domain in $\mathbb C^4$ and its connection with $μ$-synthesis problem

Abstract

We explore a domain $\mathbb F$ in $\mathbb C^4$ that has structural similarities with the hexablock $\mathbb{H} \subset \mathbb C^4$. It leads to the question if these two domains are biholomorphic. In this paper, we answer this question in negative. We provide alternative characterizations of the domain $\mathbb{F}$ and find its connection with the domains associated with $μ$-synthesis problem such as the symmetrized bidisc, the tetrablock, the pentablock and the hexablock. We also address the following question: does $\mathbb{F}$ (which is biholomorphic with $\mathbb L_4$) arise from a $μ$-synthesis problem in the same manner as $\mathbb{G}_2$ and $\mathbb{E}$ ?

Theorems & Definitions (42)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• Theorem 2.6: AglerIII, Theorem 2.1
• Theorem 2.7: Tirtha_Pal, Theorem 1.3