Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Norms of linear-fractional composition operators


Authors: P. S. Bourdon, E. E. Fry, C. Hammond and C. H. Spofford
Journal: Trans. Amer. Math. Soc. 356 (2004), 2459-2480
MSC (2000): Primary 47B33
DOI: https://doi.org/10.1090/S0002-9947-03-03374-9
Published electronically: November 25, 2003
MathSciNet review: 2048525
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a representation for the norm of the composition operator $C_\phi$ on the Hardy space $H^2$ whenever $\phi$ is a linear-fractional mapping of the form $\phi(z) = b/(cz +d)$. The representation shows that, for such mappings $\phi$, the norm of $C_\phi$ always exceeds the essential norm of $C_\phi$. Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form $\phi(z) = sz +t$ has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers $s$ and $t$, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of $C_{1/(2-z)}$ is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator $C_\phi$, for which $\Vert C_\phi\Vert> \Vert C_\phi\Vert _e$, an equation whose maximum (real) solution is $\Vert C_\phi\Vert^2$. Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.


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

  • 1. M. Appel, P. Bourdon, and J. Thrall, Norms of composition operators on the Hardy space, Experiment. Math. 5 (1996), 111-117. MR 97h:47022
  • 2. P. Avramidou and F. Jafari, On norms of composition operators on Hardy spaces, Contemp. Math. 232, Amer. Math. Soc., Providence, 1999. MR 2000b:47066
  • 3. A. Baker, Transcendental Number Theory, Cambridge University Press, New York, 1975. MR 54:10163
  • 4. P. S. Bourdon and J. H. Shapiro, Cyclic Phenomena for Composition Operators, Memoirs Amer. Math. Soc. #596, January 1997. MR 97h:47023
  • 5. P. S. Bourdon and D. Q. Retsek, Reproducing kernels and norms of composition operators, Acta Sci. Math. (Szeged) 67 (2001), 387-394. MR 2002b:47043
  • 6. P. S. Bourdon, D. Levi, S. Narayan, and J. H. Shapiro, Which linear-fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), 30-53. MR 2003m:47042
  • 7. C. C. Cowen, Composition operators on $H^2$, J. Operator Theory 9 (1983), 77-106. MR 84d:47038
  • 8. C. C. Cowen, Linear fractional composition operators, Integral Equations Operator Theory 11 (1988), 151-160. MR 89b:47044
  • 9. C. C. Cowen and T. L. Kriete, Subnormality and composition operators on $H^2$, J. Funct. Anal. 81 (1988), 298-319. MR 90c:47055
  • 10. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. MR 97i:47056
  • 11. C. C. Cowen and B. D. MacCluer, Some problems on composition operators, Contemporary Mathematics 213 (1998), 17-25. MR 99d:47029
  • 12. K. W. Dennis, Co-hyponormality of composition operators on the Hardy space, Acta Sci. Math. (Szeged) 68 (2002), 401-411. MR 2003f:47039
  • 13. P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970. MR 42:3552
  • 14. C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged), to appear.
  • 15. E. A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442-449. MR 36:6961
  • 16. D. B. Pokorny and J. E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory 45 (2003), 351-358.
  • 17. D. Q. Retsek, The Kernel Supremum Property and Norms of Composition Operators, Thesis, Washington University, 2001.
  • 18. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • 19. H. Sadraoui, Hyponormality of Toeplitz and Composition Operators, Thesis, Purdue University, 1992.
  • 20. J. H. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987), 375-404. MR 88c:47058
  • 21. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. MR 94k:47049

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B33

Retrieve articles in all journals with MSC (2000): 47B33


Additional Information

P. S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: pbourdon@wlu.edu

E. E. Fry
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: frye@wlu.edu

C. Hammond
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Address at time of publication: Department of Mathematics and Computer Science, Connecticut College, New London, Connecticut 06320
Email: cnh5u@virginia.edu, cnham@conncoll.edu

C. H. Spofford
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Address at time of publication: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: spoffordc@wlu.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03374-9
Received by editor(s): September 4, 2002
Received by editor(s) in revised form: April 27, 2003
Published electronically: November 25, 2003
Additional Notes: This research was supported in part by a grant from the National Science Foundation (DMS-0100290).
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society