  Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated
cone-edge metric satisfying a spectral Witt condition. Under these assumptions
the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the
asymptotic expansion for the resolvent trace of $\Delta$. Our method proceeds
by induction on the depth and applies in principle to a larger class of
second-order differential operators of regular-singular type, e.g., Dirac
Laplacians. Our arguments are functional analytic, do not rely on microlocal
techniques and are very explicit. The results of this paper provide a basis for
studying index theory and spectral invariants in the setting of smoothly
stratified spaces and in particular allow for the definition of
zeta-determinants and analytic torsion in this general setup.
