Extended Cesàro 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 $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X.$ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $\mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X)$ (and $\mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $C_{ψ,φ}$ via the estimates for the restrictions of $ψ$ and $φ$ 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 $\widetilde{C}_{ψ, φ}$ between spaces of Banach-valued holomorphic functions weak-associated to $\mathcal B_ν(B_X)$ and $\mathcal B_μ(B_X).$

No reviews yet.

No comments yet.