Riesz–Kolmogorov Compactness Criterion, Lorentz Convergence and Ruelle Theorem on Locally Compact Abelian Groups

Ruelle's theorem gives, for a certain class of self-adjoint operators on L2(Rn), a description of the pure point and continuous spectral subspaces of the operator in terms of bound and scattering states. We extend this characterization to arbitrary self-adjoint operators acting in L2(X), where X is an Abelian locally compact group. We replace the convergence in Cesàro mean from the standard version of Ruelle's theorem by convergence in Lorentz sense, which is sharper than any convergence in invariant mean sense. Our main tool is a description in term of position and momentum observables of relatively compact subsets of L2(X) extending the Riesz–Kolmogorov theorem.