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Compact composition operators on Besov spaces
Author(s):
Maria
Tjani
Journal:
Trans. Amer. Math. Soc.
355
(2003),
4683-4698.
MSC (2000):
Primary 47B38;
Secondary 30D45, 46E15
Posted:
July 8, 2003
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Additional information
Abstract:
We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.
References:
-
- 1.
- J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, Journal für die Reine und Angewandte Mathematik 270 (1974), 12-37. MR 50:13536
- 2.
- J. Arazy, S. D. Fisher, and J. Peetre, Mobius invariant function spaces, Journal für die Reine und Angewandte Mathematik 363 (1985), 110-145. MR 87f:30104
- 3.
- P. S. Chee, VMOA and vanishing Carleson measures, Complex Variables 25 (1994), 311-322. MR 95m:32008
- 4.
- J. Cima and W. Wogen, A Carleson theorem for the Bergman space on the ball, Journal of Operator Theory 7 (1982), 157-165. MR 83f:46022
- 5.
- J. B. Conway, Functions of One Complex Variable, second edition, Springer-Verlag, New York, 1978. MR 80c:30003
- 6.
- J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. MR 86h:46001
- 7.
- J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
- 8.
- F. Holland and D. Walsh, Growth estimates for functions in the Besov spaces
 , Proceedings of the Royal Irish Academy, Vol. 88A (1988), 1-18. MR 89m:30074 - 9.
- B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canadian Journal of Mathematics 38 (1986), 878-906. MR 87h:47048
- 10.
- P. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Transactions of the American Mathematical Society 347 (1995), 2679-2687. MR 95i:47061
- 11.
- V. V. Peller, Hankel operators of class
 and their applications. (Rational approximation, Gaussian processes, the problem of majorizing operators), English translation: Math. USSR Sbornik, Vol. 41, No. 4 (1982), 443-479. MR 82g:47022 - 12.
- L. A. Rubel and R. M. Timoney, An extremal property of the Bloch space, Proceedings of the American Mathematical Society 75 (1979), 45-49. MR 80h:30034
- 13.
- W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987. MR 88k:00002
- 14.
- J. H. Shapiro, The essential norm of a composition operator, Annals of Mathematics 125 (1987), 375-404. MR 88c:47058
- 15.
- J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. MR 94k:47049
- 16.
- M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University, 1996.
- 17.
- K. Zhu, Operator Theory on Function Spaces, Marcel Dekker, New York, 1990. MR 92c:47031
- 18.
- K. Zhu, Analytic Besov spaces, Journal of Mathematical Analysis and Applications 157 (1991), 318-336. MR 93e:30092
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Additional Information:
Maria
Tjani
Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email:
mtjani@comp.uark.edu
DOI:
10.1090/S0002-9947-03-03354-3
PII:
S 0002-9947(03)03354-3
Received by editor(s):
June 21, 1999
Received by editor(s) in revised form:
March 26, 2003
Posted:
July 8, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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