Abstract
In this article, we study the boundedness of composition operators between strong and weak version vector-valued weighted Dirichlet spaces. Some sufficient and necessary conditions for such composition operators to be bounded are obtained exactly, which are different from the scalar-valued case and agree with the characterizations for these composition operators to be Hilbert–Schmidt operators between the corresponding scalar-valued Dirichlet spaces. As a by-product, we show indirectly that these strong and weak version Dirichlet spaces are different significatively, and obtain the equivalence of some analytic properties of an analytic self-map of the unit disk.
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