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A Schwarz lemma on the polydisk

Author(s): Greg Knese
Journal: Proc. Amer. Math. Soc. 135 (2007), 2759-2768.
MSC (2000): Primary 30C80; Secondary 32A30, 47A57
Posted: March 30, 2007
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Abstract | References | Similar articles | Additional information

Abstract: We prove a generalization of the infinitesimal portion of the classical Schwarz lemma for functions from the polydisk to the disk. In particular, we describe the functions which play the role of automorphisms of the disk in this context-they turn out to be rational inner functions in the Schur-Agler class of the polydisk with an added symmetry constraint. In addition, some sufficient conditions are given for a function to be of this type.

References:

1.: J.Agler and J.E.McCarthy. Pick Interpolation and Hilbert Function Spaces. American Mathematical Society, Providence, 2002. MR 1882259 (2003b:47001)
2.: J.P.D'Angelo. Several Complex Variables and the Geometry of Real Hypersurfaces. CRC, Boca Raton, FL, 1993. MR 1224231 (94i:32022)
3.: W. Rudin Function Theory in Polydiscs. Benjamin, New York, 1969. MR 0255841 (41:501)

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Additional Information:

Greg Knese
Affiliation: Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
Email: geknese@math.wustl.edu

DOI: 10.1090/S0002-9939-07-08766-7
PII: S 0002-9939(07)08766-7
Received by editor(s): April 10, 2006
Received by editor(s) in revised form: May 1, 2006
Posted: March 30, 2007
Additional Notes: Thanks to John McCarthy for his advice at all stages of this research.
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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