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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
Published electronically: November 25, 2003
MathSciNet review: 2048525
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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.


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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