Noncommutative Berezin transforms and multivariable operator model theory ✩

In this paper, we initiate the study of a class Dmp (H) of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space H, where m 2, n 2, and p is a positive regular polynomial in n noncommutative indeterminates. These domains are defined by certain positivity conditions on p, i.e., Dmp (H) := X := (X1, . . . , Xn): (1− p)k(X,X∗) 0 for 1 k m . Each such a domain has a universal model (W1, . . . , Wn) of weighted shifts acting on the full Fock space F2(Hn) with n generators. The study of Dmp (H) is close related to the study of the weighted shifts W1, . . . , Wn, their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra An(Dmp ), the Hardy algebra F∞n (Dmp ), and the C∗-algebra C∗(W1, . . . , Wn). A good part of this paper deals with these issues. The main tool, which we introduce here, is a noncommutative Berezin type transform associated with each n-tuple of operators in Dmp (H). The study of this transform and its boundary behavior leads to Fatou type results, functional calculi, and a model theory for n-tuples of operators in Dmp (H). These results extend to noncommutative varieties Vm p,Q(H) ⊂ Dmp (H) generated by classes Q of noncommutative polynomials. When m 2, n 2, p = Z1 +···+Zn, and Q = 0, the elements of the corresponding variety Vm p,Q(H) can be seen as multivariable noncommutative analogues of Agler’s m-hypercontractions. Our results apply, in particular, when Q consists of the noncommutative polynomials ZiZj − ZjZi , i, j = 1, . . . , n. In this case, the model space is a symmetric weighted Fock space F2 s (Dmp ), which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in Cn, and the universal model is the n-tuple (Mλ1, . . . , Mλn ) of multipliers by the coordinate functions. In this particularcase, we obtain a model theory for commuting n-tuples of operators in Dmp (H), recovering several results already existent in the literature. © 2007 Elsevier Inc. All rights reserved.