ON PERTURBED POSITIVE SEMIGROUPS ON THE BANACH SPACE OF TRACE CLASS OPERATORS

Abstract

Let [Formula: see text] be the generator of a positive (i.e. leaving invariant [Formula: see text]) contraction semigroup on [Formula: see text], the space of self-adjoint trace class operators on a Hilbert space H, endowed with the trace norm, and let [Formula: see text] be a positive linear operator such that [Formula: see text], [Formula: see text]. We show that there exists a minimal positive contraction semigroup generated by some [Formula: see text] and provide a systematic study of the total mass carried by individual trajectories [Formula: see text] with non-negative initial data ρ. The analysis relies on mathematical properties of two (a priori) different extensions [Formula: see text] of the functional [Formula: see text] to [Formula: see text].