  We prove a Tb Theorem that characterizes all Calderon-Zygmund operators that
extend compactly on L^p(R^n), 1<p<\infty . The result, whose proof does not
require the property of accretivity, can be used to prove compactness of the
Double Layer Potential operator on a wide class of domains. The study also
provides conditions for boundedness of singular integral operators by means of
non-accretive testing functions.
