On the Spectrum of Two Different Fractional Operators

In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given by (−∆)u(x) := c(n, s) 2 ∫ Rn 2u(x)− u(x+ y)− u(x− y) |y|n+2s dy , x ∈ R n , where c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is Asu = ∑ i∈N ai λ s i ei , where ei , λi are the eigenfunctions and the eigenvalues of the Laplace operator −∆ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei . Aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.