Compactness of composition operators on Hardy spaces of homogeneous rooted trees

In this paper, we present a complete characterization of the compactness of arbitrary finite linear combinations of composition operators acting on Hardy spaces of homogeneous rooted trees. In particular, we obtain a characterization of compactness for linear combinations satisfying the CNC. As an application, we give an affirmative answer to a conjecture raised by Choe et al. [Trans. Amer. Math. Soc. Ser. B 9(2022), 733–756] concerning such Hardy spaces. Furthermore, we establish the phenomenon of (double) difference cancellation for linear combinations of composition operators by constructing explicit examples.