A note on differences of composition operators acting on the Hardy space
DOI:
https://doi.org/10.7146/math.scand.a-129748Abstract
When $\varphi$ and $\psi$ are holomorphic self-maps of the unit disk with bounded multiplicity, we give a compact characterization for the difference of composition operators $C_\varphi $ and $C_\psi $ acting on the Hardy space, which extends a recent result in the case of univalent maps by Choe et al.
References
Acharyya, S., and Wu, Z., Compact and Hilbert-Schmidt differences of weighted composition operators, Integral Equations Operator Theory 88 (2017), no. 4, 465–482. https://doi.org/10.1007/s00020-017-2374-x
Bourdon, P., Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003), no. 1, 228–245. https://doi.org/10.1016/S0022-247X(03)00004-0
Choe, B., Choi, K., Koo, H., and Yang, J., Differences of weighted composition operators, J. Funct. Anal. 278 (2020), no. 5, 108401, 38 pp. https://doi.org/10.1016/j.jfa.2019.108401
Choe, B., Koo, H., and Park, I., Compact differences of composition operators on the Bergman space over the ball, Potential Anal. 40 (2014), no. 1, 81–102. https://doi.org/10.1007/s11118-013-9343-z
Cowen, C., and MacCluer, B., Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995.
Dai, J., Topological structure of the set of composition operators on the weighted Bergman space, J. Math. Anal. Appl. 473 (2019), no. 1, 444–467. https://doi.org/10.1016/j.jmaa.2018.12.060
Gallardo-Gutiérrez, E., González, M., Nieminen, P., and Saksman, E., On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008), no. 3, 986–1001. https://doi.org/10.1016/j.aim.2008.06.005
Gorkin, P., Mortini, R., and Suárez, D., Homotopic composition operators on $H^{{infty}}(B^n)$, Function spaces (Edwardsville, IL, 2002), 177–188, Contemp. Math., 328, Amer. Math. Soc., Providence, RI, 2003. https://doi.org/10.1090/conm/328/05779
Koo, H., and Wang, M., Joint Carleson measure and the difference of composition operators on $A^p_{alpha}(B_n)$, J. Math. Anal. Appl. 419 (2014), no. 2, 1119–1142. https://doi.org/10.1016/j.jmaa.2014.05.037
bibitem Kri Kriete, T., and Moorhouse, J., Linear relations in the Calkin algebra for composition operators, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2915–2944. https://doi.org/10.1090/S0002-9947-07-04166-9
Lindström, M., and Saukko, E., Essential norm of weighted composition operators and difference of composition operators between standard weighted Bergman spaces, Complex Anal. Oper. Theory 9 (2015), no. 9, 1411–1432. https://doi.org/10.1007/s11785-014-0430-y
MacCluer, B., Components in the space of composition operators, Integral Equations Operator Theory 12 (1989), no. 5, 725–738. https://doi.org/10.1007/BF01194560
MacCluer, B., Ohno, S., and Zhao, R., Topological structure of the space of composition operators on $H^{infty} $, Integral Equations Operator Theory 40 (2001), no. 4, 481–494. https://doi.org/10.1007/BF01198142
MacCluer, B., and Shapiro, J., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. https://doi.org/10.4153/CJM-1986-043-4
Moorhouse, J., and Toews, C., Differences of composition operators, Trends in Banach spaces and operator theory (Memphis, TN, 2001), 207–213, Contemp. Math. 321, Amer. Math. Soc. Providence, RI, 2003. https://doi.org/10.1090/conm/321/05645
Moorhouse, J., Compact differences of composition operators, J. Funct. Anal. 219 (2005), no. 1, 70–92. https://doi.org/10.1016/j.jfa.2004.01.012
Saukko, E., Difference of composition operators between standard weighted Bergman spaces, J. Math. Anal. Appl. 381 (2011), no. 2, 789–798. https://doi.org/10.1016/j.jmaa.2011.03.058
Shapiro, J., The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. https://doi.org/10.2307/1971314
Shapiro, J., and Sundberg, C., Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no, 1, 117–152. http://projecteuclid.org/euclid.pjm/1102645610
Zhu, K., Operator theory in function spaces, Marcel Dekker, Inc., New York, 1990.
