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

 

Characterizing derivations from the disk algebra to its dual


Authors: Y. Choi and M. J. Heath
Journal: Proc. Amer. Math. Soc. 139 (2011), 1073-1080
MSC (2010): Primary 46J15; Secondary 30H10, 47B47
Published electronically: August 3, 2010
MathSciNet review: 2745657
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Abstract: We show that the space of all bounded derivations from the disk algebra into its dual can be identified with the Hardy space $ H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $ D$, we construct a finite, positive Borel measure $ \mu_D$ on the closed disk, such that $ D$ factors through $ L^2(\mu_D)$. Such a measure is known to exist, for any bounded linear map from the disk algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.


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Additional Information

Y. Choi
Affiliation: Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, QC, Canada, G1V 0A6
Address at time of publication: Department of Mathematics and Statistics, McLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada S7N 5E6
Email: y.choi.97@cantab.net

M. J. Heath
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: mheath@math.ist.utl.pt

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10520-8
PII: S 0002-9939(2010)10520-8
Keywords: Derivation, disk algebra, Hardy space
Received by editor(s): December 15, 2009
Received by editor(s) in revised form: March 30, 2010
Published electronically: August 3, 2010
Additional Notes: The second author was supported by post-doctoral grant SFRH/BPD/40762/2007 from FCT (Portugal).
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.