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)



Sampling sets and closed range composition operators on the Bloch space

Authors: Pratibha Ghatage, Dechao Zheng and Nina Zorboska
Journal: Proc. Amer. Math. Soc. 133 (2005), 1371-1377
MSC (2000): Primary 47B33
Published electronically: October 28, 2004
MathSciNet review: 2111961
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a necessary and sufficient condition for a composition operator $C_{\phi }$ on the Bloch space to have closed range. We show that when $\phi $ is univalent, it is sufficient to consider the action of $C_{\phi }$ on the set of Möbius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

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

  • 1. J. Garnett, Bounded Analytic Functions, Academic Press, 1981. MR 0628971 (83g:30037)
  • 2. P. Ghatage, J. Yan and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (2001), 2039-2044. MR 1825915 (2002a:47034)
  • 3. P. Ghatage and D. Zheng, Hyperbolic derivatives and generalized Schwarz-Pick Estimates, To appear in Proc. Amer. Math Soc..
  • 4. M. Jovovic and B. MacCluer, Composition operators on Dirichlet spaces, Acta. Sci. Math. (Szeged) 63 (1997), 229-297. MR 1459789 (98d:47067)
  • 5. D. Luecking, Inequalities on Bergman Spaces, Ill. Jour. of Math. 25 (1981), 1-11. MR 0602889 (82e:30072)
  • 6. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. MR 1273508 (95i:47061)
  • 7. C. Pommerenke, Boundary Behaviour of Conformal Maps, vol. 299, Springer-Verlag, 1992. MR 1217706 (95b:30008)
  • 8. K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113, 21-29. MR 1223222 (94g:30033)
  • 9. W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), 2331-2348. MR 1357404 (96i:47056)
  • 10. D. Stegenga and K. Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. 2 24 (1981), 243-254. MR 0631937 (82m:30036)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B33

Retrieve articles in all journals with MSC (2000): 47B33

Additional Information

Pratibha Ghatage
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Dechao Zheng
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 32740

Nina Zorboska
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T2N2
Email: zorboska@cc.umanitoba.CA

Received by editor(s): November 7, 2003
Received by editor(s) in revised form: December 30, 2003
Published electronically: October 28, 2004
Dedicated: Dedicated to Chandler Davis for his 75th birthday
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society