  Many results in the theory of Gaussian processes rely on the eigenstructure
of the covariance operator. However, eigenproblems are notoriously hard to
solve explicitly and closed form solutions are known only in a limited number
of cases. In this paper we set up a framework for the spectral analysis of the
fractional type covariance operators, corresponding to an important family of
processes, which includes the fractional Brownian motion and its noise. We
obtain accurate asymptotic approximations for the eigenvalues and the
eigenfunctions. Our results provide a key to several problems, whose solution
is long known in the standard Brownian case, but was missing in the more
general fractional setting. This includes computation of the exact limits of
$L^2$-small ball probabilities and asymptotic analysis of singularly perturbed
integral equations, arising in mathematical physics and applied probability.
