Dunford property for composition operators on $H^p$-spaces

The Dunford property $(C)$ for composition operators on $H^p$-spaces ($1<p<\mathrm{\infty }$), as well as for their adjoints, is completely characterized within the class of those induced by linear fractional transformations of the unit disc. As a consequence, it is shown that the Dunford property is stable in such a class addressing a particular instance of a question posed by Laursen and Neumann.