Composition operators on vector-valued analytic function spaces: a survey

  • Jussi Laitila Department of Biosciences, P.O. Box 65, FI-00014 University of Helsinki, Helsinki
  • Hans-Olav Tylli Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Helsinki
Keywords: analytic functions, Banach space, compactness, composition operator, weak compactness, vector-valued function


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.


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