Maps on positive operators preserving Lebesgue decompositions

Abstract

Let H be a complex Hilbert space. Denote by B(H)⁺ the set of all positive bounded linear operators on H. A bijective map ∅: B(H)⁺ → B(H)⁺ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C + D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C) + φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every such transformation φ is of the form φ(A) = SAS^∗ (A ∈ B(H)⁺) for some invertible bounded linear or conjugate-linear operator S on H.