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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Closed densely defined operators commuting with multiplications in a multiplier pair


Authors: Don Hadwin, Zhe Liu and Eric Nordgren
Journal: Proc. Amer. Math. Soc. 141 (2013), 3093-3105
MSC (2010): Primary 46B99, 47B47; Secondary 30H10, 30H15, 46L10
Published electronically: May 2, 2013
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Abstract: For a multiplier pair $ \left ( X,Y\right ) $ we study the closed densely defined operators $ T$ on $ X$ that commute with all of the multiplications by right multipliers in $ X$. We apply our general results to special cases involving $ H^{p}$, completions of $ L^{\infty }\left [ 0,1\right ] $ with respect to certain norms, and the completion of a $ II_{1}$ factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such $ T$ is a ``left multiplication''. However, we give an example of a closed densely defined operator on the Bergman space that commutes with all multiplications by $ H^{\infty }$-functions but is not a multiplication operator.


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Additional Information

Don Hadwin
Affiliation: Department of Mathematics, Kingsbury Hall, University of New Hampshire, Durham, New Hampshire 03824-3591
Email: don@math.unh.edu

Zhe Liu
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104
Email: zheliu@sas.upenn.edu

Eric Nordgren
Affiliation: Department of Mathematics, Kingsbury Hall, University of New Hampshire, Durham, New Hampshire 03824-3591
Email: ean@math.unh.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11753-3
PII: S 0002-9939(2013)11753-3
Keywords: Multiplier pair, closed operator, $II_{1}$-factor, Hardy space
Received by editor(s): November 11, 2011
Published electronically: May 2, 2013
Dedicated: Dedicated to the memory of Bill Arveson, an inspiration to us all
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.