Geodesic
Maps on positive operators preserving Lebesgue decompositions
The electronic journal of linear algebra, Tome 18 (2009), pp. 222-232.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let H be a complex Hilbert space. Denote by $B(H) + the$ set of all positive bounded linear operators on H. A bijectiv e map $\varphi : B(H) + \rightarrow B(H) + is$ 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 $\varphi (B) = \varphi (C)+\varphi (D)$ is a $\varphi (A)$-Lebesgue decomposition of $\varphi (B)$. It is proved that every such transformation $\varphi $is of the form $\varphi (A) = SAS$ * ( A $\in B(H)$ + ) for some invertible bounded linear or conjugate-linear operator S on H.
Classification : 47B49
Keywords: positive operators, Lebesgue decomposition, preservers 
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     author = {Moln\'ar, Lajos},
     title = {Maps on positive operators preserving {Lebesgue} decompositions},
     journal = {The electronic journal of linear algebra},
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     publisher = {mathdoc},
     volume = {18},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2009__18__a39/}
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Molnár, Lajos. Maps on positive operators preserving Lebesgue decompositions. The electronic journal of linear algebra, Tome 18 (2009), pp. 222-232. http://geodesic.mathdoc.fr/item/ELA_2009__18__a39/