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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Berezin transforms on noncommutative polydomains
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by Gelu Popescu PDF
Trans. Amer. Math. Soc. 368 (2016), 4357-4416 Request permission

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 $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ 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 $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$.

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Additional Information
  • Gelu Popescu
  • Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
  • MR Author ID: 234950
  • Email: gelu.popescu@utsa.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 4357-4416
  • MSC (2010): Primary 46L52, 47A56; Secondary 47A48, 47A60
  • DOI: https://doi.org/10.1090/tran/6466
  • MathSciNet review: 3453374