Ando dilations and inequalities on noncommutative domains

TL;DR

This paper develops new dilation theorems and inequalities for noncommutative domains and varieties, extending classical results like Ando's and Sz.-Nagy--Foias' theorems to broader noncommutative settings.

Contribution

It introduces intertwining dilation theorems for noncommutative regular domains and varieties, generalizing classical commutant lifting theorems and inequalities to noncommutative operator tuples.

Findings

New intertwining dilation theorems for noncommutative domains and varieties.

A Schur type representation for the Hardy algebra associated with noncommutative varieties.

Extensions of Ando's inequalities to larger classes of commuting operators.

Abstract

We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V_I. In particular, we obtain extensions of Ando's results and Agler-McCarthy's inequality for commuting contractions to larger classes of commuting operators.