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.