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 is an element of (0, 1) we consider the integral definition of the fractional Laplacian given by (-Delta)(s)u(x) := c(n, s)/2 integral(Rn) 2u(x) - u(x + y) - u(x - y)/vertical bar y vertical bar(n+2s) dy, x is an element of R-n, where c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is, A(s)(u) = Sigma(i is an element of N) a(i)lambda(s)(i)e(i), where e(i), lambda(i) are the eigenfunctions and the eigenvalues of the Laplace operator -Delta in Omega 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.