Maps on positive operators preserving Lebesgue decompositions
نویسندگان
چکیده
Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : 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.
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