Weighted composition operators between vector-valued Bloch-type spaces on the polydisk

Let $X$ and $Y$ be two complex Banach spaces, ${\mathbb{C}}^N$ the $N$-dimensional complex Euclidean space with the inner product $⟨z,w⟩=\sum _{l=1}^N{z}_l\overline{w_l}$ and ${\mathbb{D}}^N$ the unit polydisk in ${\mathbb{C}}^N$. Let $\phi$ be a holomorphic self-map of ${\mathbb{D}}^N$ and $u\in H({\mathbb{D}}^N,\mathcal{L}(X,Y))$, where $H({\mathbb{D}}^N,X)$ denotes the space of all vector-valued holomorphic functions on ${\mathbb{D}}^N$ and $\mathcal{L}(X,Y)$ denotes the space of all bounded linear operators from $X$ to $Y$. The weighted composition operator $W_{u,\phi }:H({\mathbb{D}}^N,X)\to H({\mathbb{D}}^N,Y)$ is defined by

$$\begin{array}{r}{W}_{u,\phi }f(z)=u(z)(f(\phi (z))).\end{array}$$

The bounded and compact weighted composition operators between vector-valued Bloch-type spaces on the unit polydisk are completely characterized in the paper.