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

Abstract

The hexablock is a domain arising from a special case of the $\hat{I}¼$-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 $\hat{I}¼$-synthesis problem.

32 pages, Submitted to journal