2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131980We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space $H^2$ on the unit disc. 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.16 pagesFunctional AnalysisComplex Variables47B33 (Primary), 30D55, 47B38 (Secondary)text