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

DOI:
https://doi.org/10.1090/S0002-9939-02-06596-6

Published electronically:
July 17, 2002

MathSciNet review:
1937427

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1996, C. Cowen and B. MacCluer studied a class of maps on that they called linear fractional maps. Using the tools of Krein 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 Krein 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

Email:
richman@math.purdue.edu, arichman@bucknell.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06596-6

Keywords:
Linear fractional maps,
unit ball,
Kre\u{\i}n 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