### 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 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.

Original language | English |
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Pages (from-to) | 222-232 |

Number of pages | 11 |

Journal | Electronic Journal of Linear Algebra |

Volume | 18 |

Publication status | Published - Jan 2009 |

### Fingerprint

### Keywords

- Lebesgue decomposition
- Positive operators
- Preservers

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Maps on positive operators preserving lebesgue decompositions.** / Molnár, L.

Research output: Contribution to journal › Article

*Electronic Journal of Linear Algebra*, vol. 18, pp. 222-232.

}

TY - JOUR

T1 - Maps on positive operators preserving lebesgue decompositions

AU - Molnár, L.

PY - 2009/1

Y1 - 2009/1

N2 - 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.

AB - 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.

KW - Lebesgue decomposition

KW - Positive operators

KW - Preservers

UR - http://www.scopus.com/inward/record.url?scp=65749112417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65749112417&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:65749112417

VL - 18

SP - 222

EP - 232

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

SN - 1081-3810

ER -