Weighted composition operators between vector-valued Bloch-type spaces.

Let X and Y be complex Banach spaces and 픻 be the open unit disc in the complex plane ℂ. Let φ be an analytic self-map of D and ℂ be an analytic operator-valued function from D into the space of all bounded linear operators from X to Y. The weighted composition operator : W_ψ,φ: H(픻,X) → (픻, Y) is defined by W_ψ,φ (f)(z) = ψ(z)(f (φ(z))), (z ∈ 픻,f ∈ H(픻,X)), where H(픻,X) is the space of all analytic X-valued functions on 픻. In this paper we provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators Wψ,φ between vector-valued Bloch- type spaces Bα(X) and Bβ(Y) for α, β > 0 in terms of ψ,φ, their derivatives, and the nth power φⁿ of φ. [ABSTRACT FROM AUTHOR]