Extended Ces\`aro composition operators on weak Bloch-type spaces on the unit ball of a Hilbert space

Denote by $B_X$ the unit ball of an infinite-dimensional complex Hilbert space $X.$ Let $\psi \in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $\varphi \in S(B_X)$ the set of holomorphic self-maps of $B_X.$ Let $C_{\psi, \varphi}: \mathcal B_{\nu}(B_X)$ (and $\mathcal B_{\nu,0}(B_X)$) $\to \mathcal B_{\mu}(B_X)$ (and $\mathcal B_{\mu,0}(B_X)$) be weighted extended Ces\`aro operators induced by products of the extended Ces\`aro operator $C_\varphi$ and integral operator $T_\psi.$ In this paper, we characterize the boundedness and compactness of $C_{\psi,\varphi}$ via the estimates for the restrictions of $\psi$ and $\varphi$ to a $m$-dimensional subspace of $X$ for some $m\ge2.$ Based on these we give necessary as well as sufficient conditions for the boundednees, the (weak) compactness of $\widetilde{C}_{\psi, \varphi}$ between spaces of Banach-valued holomorphic functions weak-associated to $\mathcal B_{\nu}(B_X)$ and $\mathcal B_{\mu}(B_X).$