Explicit and unique construction of tetrablock unitary dilation in a certain case

TL;DR

This paper constructs an explicit, minimal, and unique boundary normal dilation for certain operator triples associated with the tetrablock domain, advancing understanding of multivariable dilation theory.

Contribution

It provides a novel explicit construction of a boundary normal dilation for operator triples with the tetrablock as a spectral set, ensuring minimality and uniqueness under specific conditions.

Findings

Constructed explicit boundary normal dilation for tetrablock-related operator triples

Proved the dilation is minimal and unique under certain conditions

Enhanced understanding of multivariable dilation and spectral set theory

Abstract

Consider the domain $E$ in $\mathbb{C}^3$ defined by $$ E=\{(a_{11},a_{22},\text{det}A): A=\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\text{ with }\lVert A \rVert <1\}. $$ This is called the tetrablock. This paper constructs explicit boundary normal dilation for a triple $(A,B,P)$ of commuting bounded operators which has $\bar{E}$ as a spectral set. We show that the dilation is minimal and unique under a certain natural condition. As is well-known, uniqueness of minimal dilation usually does not hold good in several variables, e.g., Ando's dilation is not known to be unique. However, in the case of the tetrablock, the third component of the dilation can be chosen in such a way as to ensure uniqueness.