# Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

Autores UPV Bonet Solves, José Antonio; Gómez Collado, María Del Carmen; Jornet Casanova, David 2012 ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA

## Abstract

We investigate several properties of operator-weighted composition maps $W_{\psi, \varphi}: f \mapsto \psi (f \circ \varphi)$ on unweighted $H(\D,X)$ and weighted $H_v^{\infty}(\D,X)$ spaces of vector valued holomorphic functions on the unit disc $\D$. Here $\varphi$ is an analytic self-map of $\D$ and $\psi$ is an analytic operator-valued function on $\D$. We characterize when the operator is continuous, maps a neighbourhood into a bounded set or maps bounded sets into relatively compact sets. In this way we extend results due to Laitila and Tylli for the case of Banach valued functions. This more general setting permits us to compare the results in the unweighted and weighted case. New examples are provided, especially when the spaces $X$ and $Y$ are K\"othe echelon spaces. They show the differences between the present setting and the case of functions taking values in Banach spaces.