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Berezin transforms on noncommutative polydomains

Author: Gelu Popescu
Journal: Trans. Amer. Math. Soc. 368 (2016), 4357-4416
MSC (2010): Primary 46L52, 47A56; Secondary 47A48, 47A60
Published electronically: September 15, 2015
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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 $ {\bf D_q^m}(\mathcal {H})$ in $ 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 $ {\bf W}=\{{\bf W}_{i,j}\}$ consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra $ F^\infty ({\bf D_q^m})$ as the weakly closed algebra generated by $ \{{\bf W}_{i,j}\}$ and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in $ {\bf D_q^m}(\mathcal {H})$. It is shown that the Berezin transform is a completely isometric isomorphism between $ F^\infty ({\bf D_q^m})$ and the algebra of bounded free holomorphic functions on the radial part of $ {\bf D_q^m}(\mathcal {H})$. A characterization of the Beurling type joint invariant subspaces under $ \{{\bf 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 $ {\bf D_q^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^*$-algebra techniques, we develop a dilation theory on the noncommutative polydomain $ {\bf D_q^m}(\mathcal {H})$.

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Additional Information

Gelu Popescu
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249

Keywords: Multivariable operator theory, Berezin transform, noncommutative polydomain, free holomorphic function, characteristic function, Fock space, weighted shift, invariant subspace, functional calculus, dilation theory
Received by editor(s): November 6, 2013
Received by editor(s) in revised form: April 29, 2014
Published electronically: September 15, 2015
Additional Notes: This research was supported in part by an NSF grant
Article copyright: © Copyright 2015 American Mathematical Society