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.

Gy. P. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation Office (Grant no. K115383). Zs. Tarcsay was supported by DAAD-TEMPUS Cooperation Project "Harmonic Analysis and Extremal Problems'' (Grant no. 308015) and by Thematic Excellence Programme, Industry and Digitization Subprogramme, NRDI Office, 2019. T. Titkos was supported by the Hungarian National Research, Development and Innovation Office NKFIH (grant no. PD128374 and grant no. K115383), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-19-4-BGE-1 New National Excellence Program of the Ministry for Innovation and Technology.