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We find a lower bound for the essential norm of the difference of two composition operators acting on $H^2(B_N)$ or $A^2_s(B_N)$ ($s>-1$). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.