Partially normal composition operators relevant to weighted directed trees

We characterize properties including p-hyponormality and p-paranormality for composition operators arising from measurable transformations on weighted directed trees, in terms of a test at each node v involving the masses at nodes in a neighborhood of nodes near v. Also constructed are certain graphs E “universal” for p-hyponormality in that the neighborhood of any node in any graph yielding a p-hyponormal composition operator is a certain limit of neighborhoods in E. These results are applied to some examples with particularly regular graph structures.