On the spectrum of two different fractional operators

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given by where c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is, where e i, λ i are the eigenfunctions and the eigenvalues of the Laplace operator -Δ in Ω with homogeneous Dirichlet boundary data, while a i represents the projection of u on the direction e i. The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences. © Royal Society of Edinburgh 2014.