  We present a method for the fast computation of the eigenpairs of a bijective
positive symmetric linear operator $\mathcal{L}$. The method is based on a
combination of operator adapted wavelets (gamblets) with hierarchical subspace
correction.First, gamblets provide a raw but fast approximation of the
eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into
sparse and well-conditioned blocks. Next, the hierarchical subspace correction
method, computes the eigenpairs associated with the Galerkin restriction of
$\mathcal{L}$ to a coarse (low dimensional) gamblet subspace, and then,
corrects those eigenpairs by solving a hierarchy of linear problems in the
finer gamblet subspaces (from coarse to fine, using multigrid iteration). The
proposed algorithm is robust for the presence of multiple (a continuum of)
scales and is shown to be of near-linear complexity when $\mathcal{L}$ is an
(arbitrary local, e.g.~differential) operator mapping $\mathcal{H}^s_0(\Omega)$
to $\mathcal{H}^{-s}(\Omega)$ (e.g.~an elliptic PDE with rough coefficients).
