Bergman spaces over noncommutative domains and commutant lifting

Abstract

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H∞(D) and Sz.Nagy-Foiaş commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces.As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in Cn. In addition, we obtain Andô type dilations and inequalities on noncommutative bi-domains and varieties.