Compact differences of composition operators on weighted Dirichlet spaces

Robert Allen ; Katherine Heller ; Matthew Pons

Open Mathematics, Tome 12 (2014), p. 1040-1051 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2014-01-01

Zbl 1312.47027

EUDML-ID : urn:eudml:doc:269467

Bibliographie

[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 | Zbl 0689.47010

[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 | Zbl 0738.47001

[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 | Zbl 1043.47021

[4] Boyd D.M., Composition operators on the Bergman space, Colloq. Math., 1975/76, 34(1), 127–136 | Zbl 0322.47018

[5] Calderón A.-P., Intermediate spaces and interpolation, the complex method, Stud. Math., 1964, 24, 113–190 | Zbl 0204.13703

[6] Conway J.B., A Course in Functional Analysis, 2nd ed., Grad. Texts in Math., 96, Springer, New York, 1990 | Zbl 0706.46003

[7] Cowen C.C., MacCluer B.D., Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1995 | Zbl 0873.47017

[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 | Zbl 0787.46062

[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 | Zbl 0708.46060

[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 | Zbl 0685.47027

[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 | Zbl 0608.30050

[13] McCarthy J.E., Geometric interpolation between Hilbert spaces, Ark. Mat., 1992, 30(2), 321–330 http://dx.doi.org/10.1007/BF02384878 | Zbl 0785.46060

[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 | Zbl 1087.47032

[15] Nordgren E.A., Composition operators, Canad. J. Math., 1968, 20, 442–449 http://dx.doi.org/10.4153/CJM-1968-040-4 | Zbl 0161.34703

[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 | Zbl 1217.47050

[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 | Zbl 0927.46054

[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 | Zbl 0622.47028

[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 | Zbl 0642.47027

[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