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

 

Uniform algebras generated by holomorphic and close-to-harmonic functions


Authors: Gautam Bharali and Sushil Gorai
Journal: Proc. Amer. Math. Soc. 139 (2011), 2183-2189
MSC (2010): Primary 30E10, 32E20, 32U05, 46J15
Published electronically: November 30, 2010
MathSciNet review: 2775396
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Abstract: The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc $ \overline{\mathbb{D}}$ generated by $ z$ and $ h$, where $ h$ is a nowhere-holomorphic harmonic function on $ \mathbb{D}$ that is continuous up to $ \partial{\mathbb{D}}$, equals $ \mathcal{C}(\overline{\mathbb{D}})$. The abstract tools used by Axler and Shields make harmonicity of $ h$ an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if $ h$ is replaced by $ h+R$, where $ R$ is a non-harmonic perturbation whose Laplacian is ``small'' in a certain sense.


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Additional Information

Gautam Bharali
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
Email: bharali@math.iisc.ernet.in

Sushil Gorai
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
Email: sushil@math.iisc.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10708-6
PII: S 0002-9939(2010)10708-6
Keywords: Harmonic function, plurisubharmonic function, polynomially convex
Received by editor(s): January 20, 2010
Received by editor(s) in revised form: May 13, 2010, and June 15, 2010
Published electronically: November 30, 2010
Additional Notes: The first author is supported by the DST via the Fast Track grant SR/FTP/MS-12/2007
The second author is supported by CSIR-UGC fellowship 09/079(2063). Support is also provided by the UGC under DSA-SAP, Phase IV
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.