The root operator on invariant subspaces of the Bergman space

Abstract

For an invariant subspace $I$ of the Bergman space we study an integral operator on $I$ defined in terms of the reproducing kernel of $I$. Such an operator will be called the root operator of $I$ and its associated integral kernel will be called the root function of $I$. We obtain fundamental spectral properties of the root operator when the invariant subspace $I$ has finite index.