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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Compact Composition Operators on BMOA


Authors: P. S. Bourdon, J. A. Cima and A. L. Matheson
Journal: Trans. Amer. Math. Soc. 351 (1999), 2183-2196
MSC (1991): Primary 47B38; Secondary 30D50.
Published electronically: February 15, 1999
MathSciNet review: 1624085
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Abstract: We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk $U$ that are Poisson integrals of functions on $\partial U$, that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator $C_\phi: \operatorname{BMOA}\rightarrow\operatorname{BMOA}$ relates to the shape of $\phi(U)$ near $\partial U$, introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.


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Additional Information

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

J. A. Cima
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599
Email: cima@math.unc.edu

A. L. Matheson
Affiliation: Department of Mathematics, Lamar University, Beaumont, Texas 77710
Email: matheson@math.lamar.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02387-9
PII: S 0002-9947(99)02387-9
Received by editor(s): January 3, 1997
Received by editor(s) in revised form: March 17, 1998
Published electronically: February 15, 1999
Additional Notes: The first author was supported in part by NSF grant DMS-9401206.
The third author was supported in part by NSF grant DMS-9500835.
Article copyright: © Copyright 1999 American Mathematical Society