Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Compactness of composition operators on BMOA

Author: Wayne Smith
Journal: Proc. Amer. Math. Soc. 127 (1999), 2715-2725
MSC (1991): Primary 47B38; Secondary 30D50
Published electronically: April 15, 1999
MathSciNet review: 1600145
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [ALXZ] Aulaskari, R., Lappan, P., Xiao, J. and Zhao, R., On $\alpha $-Bloch spaces and multipliers of Dirichlet spaces, J. Math. Anal. Appl. 209 (1997), 103-121. MR 98e:30036
  • [AFP] Arazy, J., Fisher, S. D. and Peetre, J., Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. MR 87f:30104
  • [Ba] Baernstein, A., Analysis of functions of bounded mean oscillation, Aspects of contemporary complex analysis, Academic Press, New York, 1980, pp. 3-36.
  • [BCM] Bourdon, P. S., Cima, J. A. and Matheson, A. L., Compact composition operators on BMOA, preprint.
  • [ESS] Essén, M., Shea, D.F. and Stanton, C.S., A value-distribution criterion for the class $L\log L$, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), 127-150. MR 87e:30041
  • [G] Garnett, J.B., Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
  • [J] Jarchow, H., Compactness properties of composition operators, preprint.
  • [MM] Madigan, K. and Matheson, A., Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. MR 95i:47061
  • [P] Pommerenke, Ch., Boundary Behavior of Conformal Maps, Springer-Verlag, New York, Berlin, 1992.
  • [Sh1] Shapiro, J.H., The essential norm of a composition operator, Annals of Math. 127 (1987), 375-404. MR 88c:47058
  • [Sh2] Shapiro, J.H., Composition Operators and Classical Function Theory, Springer-Verlag, New York, Berlin, 1993. MR 94k:47049
  • [Sm] Smith, W., Inner functions in the hyperbolic little Bloch class, to appear, Mich. Math. J.
  • [SZ] Smith, W. and Zhao, R., Composition operators mapping into the $Q_{p}$ spaces, Analysis 17 (1997), 239-263. CMP 98:05
  • [St] Stephenson, K., Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), 565- 577. MR 81m:30029
  • [Tj] Tjani, M., Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University (1996).
  • [Ts] Tsuji, M., Potential Theory in Modern Function Theory, Chelsea, New York, 1959. MR 54:2990

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B38, 30D50

Retrieve articles in all journals with MSC (1991): 47B38, 30D50

Additional Information

Wayne Smith
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822

Keywords: Bounded mean oscillation, compact composition operators
Received by editor(s): September 22, 1997
Received by editor(s) in revised form: November 25, 1997
Published electronically: April 15, 1999
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society