We extend the classical trace formula connecting the trace of resolvent dif-
ferences of two (not necessarily self-adjoint) operators A and A0 with the logarithmic
derivative of the associated perturbation determinant from the standard case, where A
and A0 have comparable domains (i.e., one contains the other) to the case where their
square root domains are comparable. This is done for a class of positive-type operators
A, A0. We then prove an abstract result that permits to compare square root domains
and apply this to the concrete case of 2nd order elliptic partial dierential operators in
divergence form on bounded Lipschitz domains.
This is based on various joint work with S. Hofmann, R. Nichols, and M. Zinchenko.