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A Schwarz lemma on the polydisk

Author: Greg Knese
Journal: Proc. Amer. Math. Soc. 135 (2007), 2759-2768
MSC (2000): Primary 30C80; Secondary 32A30, 47A57
Published electronically: March 30, 2007
MathSciNet review: 2317950
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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 [Enhancements On Off] (What's this?)

  • Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259
  • John P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR 1224231
  • Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841

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Additional Information

Greg Knese
Affiliation: Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
MR Author ID: 813491

Received by editor(s): April 10, 2006
Received by editor(s) in revised form: May 1, 2006
Published electronically: March 30, 2007
Additional Notes: Thanks to John McCarthy for his advice at all stages of this research.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.