Operator-weighted composition operators on vector-valued analytic function spaces

Abstract

We study qualitative properties of the operator-\break weighted composition maps ${W_{\psi,\varphi}} : f\mapsto\psi(f\circ\varphi)$ on the vector-valued spaces $H^\infty_v(X)$ of $X$-valued analytic functions $f : {\mathbb{D}}\to X$, where ${\mathbb{D}}$ is the unit disk, $X$ is a complex Banach space, $\varphi$ is an analytic self-map of ${\mathbb{D}}$, $\psi$ is an analytic operator-valued function on ${\mathbb{D}}$, and $v$ is a bounded continuous weight on ${\mathbb{D}}$. Boundedness and compactness properties of ${W_{\psi,\varphi}}$ are characterized on $H^\infty_v(X)$ for infinite-dimensional $X$. It turns out that the (weak) compactness of ${W_{\psi,\varphi}}$ also involves properties of the auxiliary operator $T_\psi : x \mapsto\psi(\cdot)x$ from $X$ to $H^\infty_v(X)$, in contrast to the familiar scalar-valued setting $X = \mathbb C$.