A Reproducing Kernel Thesis for Operators on $\ell^2$--valued Bergman-type Function Spaces

Abstract: In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their $\ell^2$--valued versions. In particular, the results in this paper apply to the weighted $\ell^2$--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.