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The range of linear fractional maps on the unit ball

Author: Alexander E. Richman
Journal: Proc. Amer. Math. Soc. 131 (2003), 889-895
MSC (2000): Primary 32A10, 32A40, 47B50
Published electronically: July 17, 2002
MathSciNet review: 1937427
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Abstract: In 1996, C. Cowen and B. MacCluer studied a class of maps on $\mathbb C^N$ that they called linear fractional maps. Using the tools of Kreĭn spaces, it can be shown that a linear fractional map is a self-map of the ball if and only if an associated matrix is a multiple of a Kreĭn contraction. In this paper, we extend this result by specifying this multiple in terms of eigenvalues and eigenvectors of this matrix, creating an easily verified condition in almost all cases. In the remaining cases, the best possible results depending on fixed point and boundary behavior are given.

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Additional Information

Alexander E. Richman
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: as of August 11, 2002: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Keywords: Linear fractional maps, unit ball, Kreĭn space
Received by editor(s): September 12, 2001
Received by editor(s) in revised form: October 19, 2001
Published electronically: July 17, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society