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 ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.

The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.