Sampling sets and closed range composition operators on the Bloch space

Ghatage, Pratibha; Zheng, Dechao; Zorboska, Nina

doi:10.1090/S0002-9939-04-07646-4

Sampling sets and closed range composition operators on the Bloch space
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by Pratibha Ghatage, Dechao Zheng and Nina Zorboska PDF

Proc. Amer. Math. Soc. 133 (2005), 1371-1377 Request permission

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

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
Pratibha Ghatage, Jun Yan, and Dechao Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2039–2044. MR 1825915, DOI 10.1090/S0002-9939-00-05771-3
P. Ghatage and D. Zheng, Hyperbolic derivatives and generalized Schwarz-Pick Estimates, To appear in Proc. Amer. Math Soc..
Mirjana Jovović and Barbara MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1997), no. 1-2, 229–247. MR 1459789
Daniel H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11. MR 602889
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
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Kristian Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), no. 1, 21–39. MR 1223222, DOI 10.1007/BF01244300
Wayne Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2331–2348. MR 1357404, DOI 10.1090/S0002-9947-96-01647-9
David A. Stegenga and Kenneth Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), no. 2, 243–254. MR 631937, DOI 10.1112/jlms/s2-24.2.243