Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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 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.

References [Enhancements On Off] (What's this?)

  • 1. 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
  • 2. C. Bisi and F. Bracci, Linear fractional maps of the unit ball: A geometric study, preprint, 2000.
  • 3. János Bognár, Indefinite inner product spaces, Springer-Verlag, New York-Heidelberg, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. MR 0467261
  • 4. Filippo Bracci, On the geometry at the boundary of holomorphic self-maps of the unit ball of 𝐶ⁿ, Complex Variables Theory Appl. 38 (1999), no. 3, 221–241. MR 1694318
  • 5. 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
  • 6. -, Schroeder's equation in several variables, preprint, August 1999.
  • 7. 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
  • 8. D. Crosby, A breakdown of linear fractional maps of the ball, unpublished notes from research as an undergraduate, 1996.
  • 9. 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
  • 10. 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
  • 11. M. G. Kre{\u{\i}}\kern.15emn and Ju. L. Smul'jan, Plus-operators in a space with indefinite metric, Amer. Math. Soc. Transl. (2) 85 (1969), 93-113.
  • 12. V. P. Potapov, Linear fractional transformations of matrices, Amer. Math. Soc. Transl. (2) 138 (1988), 21-35.
  • 13. Binyamin Schwarz and Abraham Zaks, Non-Euclidean motions in projective matrix spaces, Linear Algebra Appl. 137/138 (1990), 351–361. MR 1067682, 10.1016/0024-3795(90)90134-X
  • 14. Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406
  • 15. Ju. L. Šmul′jan, General linear-fractional transformations of operator spheres, Sibirsk. Mat. Ž. 19 (1978), no. 2, 418–425, 480 (Russian). MR 0493458

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32A10, 32A40, 47B50

Retrieve articles in all journals with MSC (2000): 32A10, 32A40, 47B50

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, 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