Compactness of composition operators on BMOA

Smith, Wayne

doi:10.1090/S0002-9939-99-04856-X

Compactness of composition operators on BMOA
HTML articles powered by AMS MathViewer

by Wayne Smith

Proc. Amer. Math. Soc. 127 (1999), 2715-2725

DOI: https://doi.org/10.1090/S0002-9939-99-04856-X

Published electronically: April 15, 1999

PDF | Request permission

Abstract:

A function theoretic characterization is given of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation on the unit circle. When the symbol of the composition operator is univalent, compactness on BMOA is shown to be equivalent to compactness on the Bloch space, and a characterization in terms of the geometry of the image of the disk under the symbol of the operator results.

References

Rauno Aulaskari, Peter Lappan, Jie Xiao, and Ruhan Zhao, On $\alpha$-Bloch spaces and multipliers of Dirichlet spaces, J. Math. Anal. Appl. 209 (1997), no. 1, 103–121. MR 1444515, DOI 10.1006/jmaa.1997.5345
J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145. MR 814017, DOI 10.1007/BFb0078341
Baernstein, A., Analysis of functions of bounded mean oscillation, Aspects of contemporary complex analysis, Academic Press, New York, 1980, pp. 3–36.
Bourdon, P. S., Cima, J. A. and Matheson, A. L., Compact composition operators on BMOA, preprint.
M. Essén, D. F. Shea, and C. S. Stanton, A value-distribution criterion for the class $L\,\textrm {log}\,L$, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 127–150 (English, with French summary). MR 812321, DOI 10.5802/aif.1030
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Jarchow, H., Compactness properties of composition operators, preprint.
Kevin Madigan and Alec Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679–2687. MR 1273508, DOI 10.1090/S0002-9947-1995-1273508-X
Pommerenke, Ch., Boundary Behavior of Conformal Maps, Springer-Verlag, New York, Berlin, 1992.
Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
Smith, W., Inner functions in the hyperbolic little Bloch class, to appear, Mich. Math. J.
Smith, W. and Zhao, R., Composition operators mapping into the $Q_{p}$ spaces, Analysis 17 (1997), 239–263.
Kenneth Stephenson, Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), no. 2, 565–577. MR 586736, DOI 10.1090/S0002-9947-1980-0586736-2
Tjani, M., Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University (1996).
M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898