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

Abstract

Consider the domain E in C³ defined by E={(a₁₁,a₂₂,detA) : A = (a₁₁,a₁₂ a₂₁,a₂₂) with ||A||<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 Ē 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 known to be not unique, see Li and Timotin (J Funct Anal 154:1–16, 1998). However, in the case of the tetrablock, the third component of the dilation can be chosen in such a way as to ensure uniqueness.