2025/06/18 by Indranil Biswas, Biswas, Indranil, Sourav Pal +3
Computer Science · #Complex Variables (math.CV) #Distributed and Parallel Computing Systems #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2506.15149
openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We introduce a domain named hexablock in \mathbb C4 and show that its origin is a special case of μ-synthesis in M2(\mathbb C), more precisely the μE-unit ball with respect to the linear subspace E consisting of 2 × 2 upper triangular matrices. The hexablock is denoted by \mathbb H and is defined by ℍ=\(a, x1, x2, x3) ∈ ℂ × 𝔼 \vert supz1, z2 ∈ \mathbb D|(a√((1-|z1|2)(1-|z2|2)))/(1-x1z1-x2z2+x3z1z2)| lt;1\, where 𝔼 is the tetrablock, another domain in \mathbb C3 associated with a different case of μ-synthesis, and is given by 𝔼=\(x1, x2, x3) ∈ ℂ3 : 1-x1z1-x2z2+x3z1z2 ≠ 0 for all z1, z2 ∈ \mathbb D\. We show that two other objects in \mathbb C4 namely, the μ-hexablock \mathbb Hμ and the normed hexablock \mathbb HN naturally arise in the μE-unit ball and the norm unit ball of M2(\mathbb C), respectively and pave the way to reach the domain \mathbb H. A set of independent characterizations for the points in \mathbb Hμ, \mathbb HN and \mathbb H are obtained. Geometric and function theoretic aspects of \mathbb H are studied and its connections with the popular domains such as symmetrized bidisc \mathbb G2, tetrablock \mathbb E and pentablock \mathbb P are explored.