Function theory of the hexablock and applications to the tetrablock and Euclidean biball

2026/08/01 by Sourav Pal, Nitin Tomar
Mathematics · #math.CV #math.FA #math.OA

paper · pdf

34 Pages, This is just a primary draft, a revised version will appear soon

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

The realization, interpolation, extension and Toeplitz corona problems are amongst the central themes in function theory of a domain in ℂ^d. The present work addresses these four problems for the hexablock ℍ, a domain in ℂ⁴ arising in connection with a special case of μ-synthesis in H^∞ control theory. We determine Schur-Agler class SA(ℍ) for ℍ and find a realization formula for \mathbb H. With the help of this realization formula we state and prove interpolation, extension and Toeplitz corona theorems for the hexablock. As an application, we obtain analogous theorems for the Euclidean unit ball in ℂ². Moreover, we recover the existing same results for the tetrablock \mathbb E, another domain associated with the μ-synthesis, as consequences of the hexablock theory and in terms of the Schur-Agler class SA(ℍ) and admissible kernels on ℍ.

Citations

• Function Theory on Tetrablock: Realization, Interpolation, Extension and Toeplitz Corona Theorem
• Commutant lifting, interpolation, and perturbations on the polydisc
• Interpolating sequences and the Toeplitz corona theorem on the symmetrized bidisk
• Commutant lifting and Nevanlinna-Pick interpolation in several variables
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• Nevanlinna-Pick interpolation problem in the ball
• Three-point Nevanlinna Pick problem in the polydisc
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• Transfer-function realization for multipliers of the Arveson space
• Interpolation in Semigroupoid Algebras
• Subalgebras of C*-algebras III: multivariable operator theory