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2014 | 12 | 7 | 1040-1051
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Compact differences of composition operators on weighted Dirichlet spaces

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Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.
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Bibliografia
  • [1] Barnes B.A., Continuity properties of the spectrum of operators on Lebesgue spaces, Proc. Amer. Math. Soc., 1989, 106(2), 415–421 http://dx.doi.org/10.1090/S0002-9939-1989-0969515-7
  • [2] Barnes B.A., Interpolation of spectrum of bounded operators on Lebesgue spaces, Rocky Mountain J. Math., 1990, 20(2), 359–378 http://dx.doi.org/10.1216/rmjm/1181073112
  • [3] Bourdon P.S., Components of linear-fractional composition operators, J. Math. Anal. Appl., 2003, 279(1), 228–245 http://dx.doi.org/10.1016/S0022-247X(03)00004-0
  • [4] Boyd D.M., Composition operators on the Bergman space, Colloq. Math., 1975/76, 34(1), 127–136
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  • [6] Conway J.B., A Course in Functional Analysis, 2nd ed., Grad. Texts in Math., 96, Springer, New York, 1990
  • [7] Cowen C.C., MacCluer B.D., Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1995
  • [8] Cwikel M., Real and complex interpolation and extrapolation of compact operators, Duke Math. J., 1992, 65(2), 333–343 http://dx.doi.org/10.1215/S0012-7094-92-06514-8
  • [9] Heller K.C., Composition Operators on \(S^2 \left( \mathbb{D} \right)\) , PhD thesis, University of Virginia, Charlottesville, 2010
  • [10] Herrero D.A., Saxe-Webb K., Spectral continuity in complex interpolation, Math. Balkanica (N.S.), 1989, 3(3–4), 325–336
  • [11] MacCluer B.D., Components in the space of composition operators, Integral Equations Operator Theory, 1989, 12(5), 725–738 http://dx.doi.org/10.1007/BF01194560
  • [12] MacCluer B.D., Shapiro J.H., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math., 1989, 38(4), 878–906
  • [13] McCarthy J.E., Geometric interpolation between Hilbert spaces, Ark. Mat., 1992, 30(2), 321–330 http://dx.doi.org/10.1007/BF02384878
  • [14] Moorhouse J., Compact differences of composition operators, J. Funct. Anal., 2005, 219(1), 70–92 http://dx.doi.org/10.1016/j.jfa.2004.01.012
  • [15] Nordgren E.A., Composition operators, Canad. J. Math., 1968, 20, 442–449 http://dx.doi.org/10.4153/CJM-1968-040-4
  • [16] Pons M.A., The spectrum of a composition operator and Calderón’s complex interpolation, In: Topics in Operator Theory, 1, Oper. Theory Adv. Appl., 202, Birkhäuser, Basel, 2010, 451–467
  • [17] Saxe K., Compactness-like operator properties preserved by complex interpolation, Ark. Mat., 1997, 35(2), 353–362 http://dx.doi.org/10.1007/BF02559974
  • [18] Shapiro J.H., Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc., 1987, 100(1), 49–57 http://dx.doi.org/10.1090/S0002-9939-1987-0883400-9
  • [19] Shapiro J.H., The essential norm of a composition operator, Ann. Math., 1987, 125(2), 375–404 http://dx.doi.org/10.2307/1971314
  • [20] Shapiro J.H., Composition Operators and Classical Function Theory, Universitext Tracts Math., Springer, New York, 1993 http://dx.doi.org/10.1007/978-1-4612-0887-7
  • [21] Shapiro J.H., Taylor P.D., Compact, nuclear, and Hilbert-Schmidt composition operators on H 2, Indiana Univ. Math. J., 1973/74, 23, 471–496 http://dx.doi.org/10.1512/iumj.1974.23.23041
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bwmeta1.element.doi-10_2478_s11533-013-0397-3
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