Andô dilations and inequalities on non-commutative domains

Abstract

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟_f and non-commutative varieties 𝒱_J in B(𝓗)ⁿ, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's 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 (respectively, 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 Andô-type dilations and inequalities for bi-domains 𝒟_f ×_c 𝒟_g consisting of all pairs (X,Y ) of tuples X := (X₁,…, X_n1) ∊ 𝒟_f and Y := (Y₁,…, Y_n2) ∊ 𝒟_g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n₁ = n₂ = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱_J1×c 𝒱_J2, where 𝒱_J1 and 𝒱_J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n₁ = n₂ = 1 are also discussed.