  In this paper we study the eigenvalue problems for a nonlocal operator of
order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that
there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first
one is positive, simple, isolated and has a positive and bounded associated
eigenfunction. For the first eigenvalue we also analyze the limits as $p\to
\infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo
infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a
local operator of $p-$Laplacian type). To perform this study we have to
introduce anisotropic fractional Sobolev spaces and prove some of their
properties.
