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Weighted composition operators on weak holomorphic spaces and application to weak Bloch-type spaces on the unit ball of a Hilbert space

Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some characterizations of the separating subspace $ W. $ As an application, via the estimates for the restrictions of $ ψ$ and $ φ$ to a $ m$-dimensional subspace of $ X $ for some $ m\ge2, $ we characterize the properties mentioned above of $ W_{ψ,φ} $ on Bloch-type spaces $ \mathcal B_μ(B_X) $ of holomorphic functions on the unit ball $ B_X $ of an infinite-dimensional Hilbert space as well as their the associated spaces $W\mathcal B_μ(B_X,Y), $ where $ μ$ is a normal weight on $ B_X. $