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 $W_{\psi,\varphi}: f\mapsto \psi\cdot(f \circ \varphi)$ on $E$ and $\widetilde{W}_{\psi,\varphi}: g\mapsto \psi\cdot(g \circ \varphi)$ on $WE(Y)$ via some characterizations of the separating subspace $W.$ As an application, via the estimates for the restrictions of $\psi$ and $\varphi$ to a $m$-dimensional subspace of $X$ for some $m\ge2,$ we characterize the properties mentioned above of $W_{\psi,\varphi}$ on Bloch-type spaces $\mathcal B_\mu(B_X)$ of holomorphic functions on the unit ball $B_X$ of an infinite-dimensional Hilbert space as well as their the associated spaces $W\mathcal B_\mu(B_X,Y),$ where $\mu$ is a normal weight on $B_X.$