Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 3453374
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ 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 $\mathbf {W}=\{\mathbf {W}_{i,j}\}$ consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ as the weakly closed algebra generated by $\{\mathbf {W}_{i,j}\}$ and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. It is shown that the Berezin transform is a completely isometric isomorphism between $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ and the algebra of bounded free holomorphic functions on the radial part of $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. A characterization of the Beurling type joint invariant subspaces under $\{\mathbf {W}_{i,j}\}$ 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 $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ 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 $C^*$-algebra techniques, we develop a dilation theory on the noncommutative polydomain $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L52, 47A56, 47A48, 47A60

Retrieve articles in all journals with MSC (2010): 46L52, 47A56, 47A48, 47A60

Additional Information

Gelu Popescu
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
MR Author ID: 234950

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