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

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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.

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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