Maps on positive operators preserving lebesgue decompositions

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)222-232
Number of pages11
JournalElectronic Journal of Linear Algebra
Volume18
Publication statusPublished - Jan 2009

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Positive Operator
Henri Léon Lebésgue
Decompose
Positive Linear Operators
Quadruple
Bijective
Bounded Linear Operator
Invertible
Linear Operator
Hilbert space
If and only if
Denote

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.

In: Electronic Journal of Linear Algebra, Vol. 18, 01.2009, p. 222-232.

Research output: Contribution to journalArticle

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