Ando dilations and inequalities on noncommutative varieties

TL;DR

This paper develops generalized dilation theorems and inequalities for noncommutative varieties, extending classical results like Sz.-Nagy and Ando's theorems to a broader noncommutative multivariable setting.

Contribution

It introduces dilation results that unify and extend classical theorems to noncommutative varieties and Poisson kernels, providing sharper inequalities for contractions.

Findings

Generalized dilation theorems for noncommutative varieties

Sharper inequalities than classical Ando and Agler-McCarthy inequalities

Extension of results to bi-ball and broader noncommutative settings

Abstract

We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0. Our results extend to the bi-ball and a large class of noncommutative varieties.