  In this paper we consider the cone of all positive, bounded operators acting
on an infinite dimensional, complex Hilbert space, and examine bijective maps
that preserve absolute continuity in both directions. It turns out that these
maps are exactly those that preserve singularity in both directions. Moreover,
in some weak sense, such maps are always induced by bounded, invertible,
linear- or conjugate linear operators of the underlying Hilbert space. Our
result gives a possible generalization of a recent theorem of Molnar which
characterizes maps on the positive cone that preserve the Lebesgue
decomposition of operators.
