Berezin transforms on noncommutative polydomains

Abstract

This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) in B ( H ) n B(\mathcal {H})^n . An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model W = { W i , j } \mathbf {W}=\{\mathbf {W}_{i,j}\} consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra F ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) as the weakly closed algebra generated by { W i , j } \{\mathbf {W}_{i,j}\} and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . It is shown that the Berezin transform is a completely isometric isomorphism between F ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) and the algebra of bounded free holomorphic functions on the radial part of D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . A characterization of the Beurling type joint invariant subspaces under { W i , j } \{\mathbf {W}_{i,j}\} is also provided. It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy–Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and C ∗ C^* -algebra techniques, we develop a dilation theory on the noncommutative polydomain D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) .