In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Ho¨lder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for reðTÞ, the essential spectral radius of T, and establish a formula for reðTÞ whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.
