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Weakly compact composition operators on analytic vector-valued function spaces. (English) Zbl 1075.47506
It is established that if X is a Banach space, then the composition operator on X-valued Hardy spaces, weighted Bergman spaces, and on Bloch spaces is weakly compact, respectively Rosenthal (i.e., strictly cosingular) if and only if both the identity operator on X and the corresponding composition operator on the analogous scalar valued spaces have this same property. The final section of the paper contains some results for ‘general vector-valued spaces’, where the range space is defined in terms of a Banach space of analytic functions on the unit disc whose closed unit ball is compact in the compact open topology.
47B33Composition operators
47B10Operators belonging to operator ideals
46E40Spaces of vector- and operator-valued functions
46E15Banach spaces of continuous, differentiable or analytic functions