Composition operators on weighted Dirichlet spaces

Zorboska, Nina

doi:10.1090/S0002-9939-98-04266-X

Composition operators on weighted Dirichlet spaces
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by Nina Zorboska

Proc. Amer. Math. Soc. 126 (1998), 2013-2023

DOI: https://doi.org/10.1090/S0002-9939-98-04266-X

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Abstract:

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

References

C.C.Cowen and B.D.MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
Daniel H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11. MR 602889
Daniel H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), no. 2, 345–368. MR 899655, DOI 10.1016/0022-1236(87)90072-3
Daniel H. Luecking and Ke He Zhu, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (1992), no. 5, 1127–1145. MR 1183534, DOI 10.2307/2374892
Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4
D.J. Newman, H.S. Shapiro, Embedding theorems for weighted classes of harmonic and analytic functions, Michigan Math. J. 9 (1962), 249-255
V.L.Oleinik, Embedding theorems for weighted classes of harmonic and analytic functions, Journal of Soviet Mathematics 9 (1978), 228-243.
Joel H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 1, 49–57. MR 883400, DOI 10.1090/S0002-9939-1987-0883400-9
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
J.H. Shapiro, Private communication
Wayne Smith and David A. Stegenga, A geometric characterization of Hölder domains, J. London Math. Soc. (2) 35 (1987), no. 3, 471–480. MR 889369, DOI 10.1112/jlms/s2-35.3.471
Nina Zorboska, Composition operators with closed range, Trans. Amer. Math. Soc. 344 (1994), no. 2, 791–801. MR 1236226, DOI 10.1090/S0002-9947-1994-1236226-9