Extended Cesàro composition operators on weak Bloch-type spaces on the unit ball of a Hilbert space

Abstract

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 $B_X,$ $\phi \in S(B_X)$ the set of holomorphic self-maps of $B_X.$ Let $C_{\psi ,\phi }:{\mathcal{B}}_{\nu }(B_X)$ (and ${\mathcal{B}}_{\nu ,0}(B_X)$) $\to {\mathcal{B}}_{\mu }(B_X)$ (and ${\mathcal{B}}_{\mu ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $C_{\phi }$ and integral operator $T_{\psi }.$ In this paper, we characterize the boundedness and compactness of $C_{\psi ,\phi }$ via the estimates for the restrictions of $\psi$ and $\phi$ to a $m$-dimensional subspace of $X$ for some $m\ge 2.$ Based on these we give necessary as well as sufficient conditions for the boundednees, the (weak) compactness of ${\tilde{C}}_{\psi ,\phi }$ between spaces of Banach-valued holomorphic functions weak-associated to ${\mathcal{B}}_{\nu }(B_X)$ and ${\mathcal{B}}_{\mu }(B_X).$

arXiv admin note: substantial text overlap with arXiv:2203.03080