Andô dilations and inequalities on noncommutative varieties

Abstract

Andô proved a dilation result that implies his celebrated inequality which says that if T1 and T2 are commuting contractions on a Hilbert space, then for any polynomial p in two variables,‖p(T1,T2)‖≤‖p‖D2, where D2 is the bidisk in C2. The main goal of the present paper is to find analogues of Andô's results for the elements of the bi-ball Pn1,n2 which consists of all pairs (X,Y) of row contractions X:=(X1,…,Xn1) and Y:=(Y1,…,Yn2) which commute, i.e. each entry of X commutes with each entry of Y. The results are obtained in a more general setting, namely, when X and Y belong to noncommutative varieties V1 and V2 determined by row contractions subject to constraints such asq(X1,…,Xn1)=0andr(Y1,…,Yn2)=0,q∈P,r∈R, respectively, where P and R are sets of noncommutative polynomials. We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Andô's dilation theorem for commuting contractions, Sz.-Nagy–Foiaş commutant lifting theorem, and Schur's representation for the unit ball of H∞, in the framework of noncommutative varieties and Poisson kernels on Fock spaces. This leads to one of the main results of the paper, an Andô type inequality on noncommutative varieties, which, in the particular case when n1=n2=1 and T1 and T2 are commuting contractive matrices with spectrum in the open unit disk D:={z∈C:|z|<1}, takes the form‖p(T1,T2)‖≤min⁡{‖p(B1⊗ICd1,φ1(B1))‖,‖p(φ2(B2),B2⊗ICd2)‖}, where (B1⊗ICd1,φ1(B1)) and (φ2(B2),B2⊗ICd2) are analytic dilations of (T1,T2) while B1 and B2 are the universal models associated with T1 and T2, respectively. In this setting, the inequality is sharper than Andô's inequality and Agler–McCarthy's inequality. We obtain more general inequalities for arbitrary commuting contractive matrices and improve Andô's inequality for commuting contractions when at least one of them is of class C0.We prove that there is a universal model (S⊗Iℓ2,φ(S)), where S is the unilateral shift and φ(S) is an isometric analytic Toeplitz operator on H2(D)⊗ℓ2, such that‖[prs(T1,T2)]k‖≤‖[prs(S⊗Iℓ2,φ(S))]k‖, for any commuting contractions T1 and T2 on Hilbert spaces, any k×k matrix [prs]k of polynomials in C[z,w], and any k∈N. Analogues of this result for the bi-ball Pn1,n2 and for a class of noncommutative varieties are also considered.