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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a representation for the norm of the composition operator on the Hardy space whenever is a linear-fractional mapping of the form . The representation shows that, for such mappings , the norm of always exceeds the essential norm of . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers and , Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of 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 , for which , an equation whose maximum (real) solution is . 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.

**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*, 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*, 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**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**

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