Noncommutative Berezin transforms and multivariable operator model theory

Abstract

In this paper, we initiate the study of a class D p m ( 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., D p m ( H ) : = { X : = ( X 1 , … , X n ) : ( 1 − p ) k ( X , X ∗ ) ⩾ 0 for 1 ⩽ k ⩽ m } . Each such a domain has a universal model ( W 1 , … , W n ) of weighted shifts acting on the full Fock space F 2 ( H n ) with n generators. The study of D p m ( H ) is close related to the study of the weighted shifts W 1 , … , W n , their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra A n ( D p m ) , the Hardy algebra F n ∞ ( D p m ) , and the C ∗ -algebra C ∗ ( W 1 , … , W n ) . 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 D p m ( 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 D p m ( H ) . These results extend to noncommutative varieties V p , Q m ( H ) ⊂ D p m ( H ) generated by classes Q of noncommutative polynomials. When m ⩾ 2 , n ⩾ 2 , p = Z 1 + ⋯ + Z n , and Q = 0 , the elements of the corresponding variety V p , Q m ( 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 Z i Z j − Z j Z i , i , j = 1 , … , n . In this case, the model space is a symmetric weighted Fock space F s 2 ( D p m ) , which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in C n , and the universal model is the n-tuple ( M λ 1 , … , M λ n ) of multipliers by the coordinate functions. In this particular case, we obtain a model theory for commuting n-tuples of operators in D p m ( H ) , recovering several results already existent in the literature.