| In his work, J. H. Shapiro provided an integral formula for the Nevanlinna counting function and used it to prove many results for composition operators on the Hardy space of the disk. We derive an integral formula for a counting function in the upper half-plane and use it to provide a function theoretic proof that composition operators are bounded above on the Hardy space of the upper half-plane. We also derive a new tool, the reproducing kernel thesis, to show that composition operators have closed range on the Hardy space of the disk. With it we are able to provide a direct geometric equivalence between the criterion of Cima, Thomson, and Wogen and the one of Zorboska for composition operators to have closed range on the Hardy space of the disk. |
