Operators associated with a domain in $\mathbb C^4$ and applications

Abstract

The hexablock is a domain arising from a special case of the $μ$-synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is called a hexablock-contraction or simply $\mathbb H$-contraction. We characterize the unitaries and isometries associated with $\mathbb H$-contractions. Two different types of dilation results for $\mathbb H$-contractions are obtained. We find connection of this theory with the operators associated with the symmetrized bidisc and tetrablock, two other domains related to the $μ$-synthesis problem.