In this paper we deal with two non-local operators that are both well known andwidely studied in the literature in connection with elliptic problems offractional type. More precisely, for a fixed s ∈(0,1) we consider the integral definition of the fractionalLaplacian given by

where c(n, s) is a positive normalizingconstant, and another fractional operator obtained via aspectral definition, that is,

where ei, λi are the eigenfunctions and theeigenvalues of the Laplace operator −Δ inΩ with homogeneous Dirichlet boundary data,while ai represents the projection of u on the directionei.

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