Atoms and associated spectral properties forpositive operators on Lp

Inspired by Schwartz (1961) and Jang- Lewis and Victory (1993), who study generalizations of triangularizations of matrices to operators, we shall give equivalent definitions of atoms (maximal irreducible sets) for positive operators on Lebesgue spaces.We also characterize positive power compact operators having a unique nonzero atom which appear as a natural generalization of irreducible operators and are also considered in epidemiological models.Using the different characterizations of atoms, we also provide a short proof for the representation of the ascent of a positive power compact operator as the maximal length in the graph of critical atoms.