We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H2. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures. © 2008 Elsevier Inc. All rights reserved.
Gallardo-Gutiérrez, E. A., González, M. J., Nieminen, P. J., & Saksman, E. (2008). On the connected component of compact composition operators on the Hardy space. Advances in Mathematics, 219(3), 986–1001. https://doi.org/10.1016/j.aim.2008.06.005