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 $\mathbb C^N$ 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.

- Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo,
*Schur functions, operator colligations, and reproducing kernel Pontryagin spaces*, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR**1465432** - C. Bisi and F. Bracci,
*Linear fractional maps of the unit ball: A geometric study*, preprint, 2000. - János Bognár,
*Indefinite inner product spaces*, Springer-Verlag, New York-Heidelberg, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. MR**0467261** - Filippo Bracci,
*On the geometry at the boundary of holomorphic self-maps of the unit ball of ${\bf C}^n$*, Complex Variables Theory Appl.**38**(1999), no. 3, 221–241. MR**1694318**, DOI https://doi.org/10.1080/17476939908815166 - Carl C. Cowen and Barbara D. MacCluer,
*Composition operators on spaces of analytic functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR**1397026** - ---,
*Schroeder’s equation in several variables*, preprint, August 1999. - Carl C. Cowen and Barbara D. MacCluer,
*Linear fractional maps of the ball and their composition operators*, Acta Sci. Math. (Szeged)**66**(2000), no. 1-2, 351–376. MR**1768872** - D. Crosby,
*A breakdown of linear fractional maps of the ball*, unpublished notes from research as an undergraduate, 1996. - Michael A. Dritschel and James Rovnyak,
*Extension theorems for contraction operators on Kreĭn spaces*, Extension and interpolation of linear operators and matrix functions, Oper. Theory Adv. Appl., vol. 47, Birkhäuser, Basel, 1990, pp. 221–305. MR**1120277** - J. William Helton, Joseph A. Ball, Charles R. Johnson, and John N. Palmer,
*Operator theory, analytic functions, matrices, and electrical engineering*, CBMS Regional Conference Series in Mathematics, vol. 68, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1987. MR**896034** - M. G. Kreĭn and Ju. L. Šmul′jan,
*Plus-operators in a space with indefinite metric*, Amer. Math. Soc. Transl. (2)**85**(1969), 93–113. - V. P. Potapov,
*Linear fractional transformations of matrices*, Amer. Math. Soc. Transl. (2)**138**(1988), 21–35. - Binyamin Schwarz and Abraham Zaks,
*Non-Euclidean motions in projective matrix spaces*, Linear Algebra Appl.**137/138**(1990), 351–361. MR**1067682**, DOI https://doi.org/10.1016/0024-3795%2890%2990134-X - Joel H. Shapiro,
*Composition operators and classical function theory*, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR**1237406** - Ju. L. Šmul′jan,
*General linear-fractional transformations of operator spheres*, Sibirsk. Mat. Ž.**19**(1978), no. 2, 418–425, 480 (Russian). MR**0493458**

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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

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