### Abstract

Let H be a Hilbert space and let A and B be standard *-operator algebras on H. Denote by A_{s} and B_{s} the set of all self-adjoint operators in A and B, respectively. Assume that M : A_{s} → B_{s} and M^{*} : B_{s} → A_{s} are surjective maps such that M (A M^{*} (B) A) = M (A) B M (A) and M^{*} (B M (A) B) = M^{*} (B) A M^{*} (B) for every pair A ∈ A_{s}, B ∈ B_{s}. Then there exist an invertible bounded linear or conjugate-linear operator T : H → H and a constant c ∈ {- 1, 1} such that M (A) = c T A T^{*}, A ∈ A_{s}, and M^{*} (B) = c T^{*} B T, B ∈ B_{s}.

Original language | English |
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Pages (from-to) | 302-309 |

Number of pages | 8 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 327 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2007 |

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### Keywords

- Elementary operator of length one
- Self-adjoint operator
- Standard *-operator algebra

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*327*(1), 302-309. https://doi.org/10.1016/j.jmaa.2006.04.039

**Elementary operators on self-adjoint operators.** / Molnár, L.; Šemrl, Peter.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 327, no. 1, pp. 302-309. https://doi.org/10.1016/j.jmaa.2006.04.039

}

TY - JOUR

T1 - Elementary operators on self-adjoint operators

AU - Molnár, L.

AU - Šemrl, Peter

PY - 2007/3/1

Y1 - 2007/3/1

N2 - Let H be a Hilbert space and let A and B be standard *-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that M : As → Bs and M* : Bs → As are surjective maps such that M (A M* (B) A) = M (A) B M (A) and M* (B M (A) B) = M* (B) A M* (B) for every pair A ∈ As, B ∈ Bs. Then there exist an invertible bounded linear or conjugate-linear operator T : H → H and a constant c ∈ {- 1, 1} such that M (A) = c T A T*, A ∈ As, and M* (B) = c T* B T, B ∈ Bs.

AB - Let H be a Hilbert space and let A and B be standard *-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that M : As → Bs and M* : Bs → As are surjective maps such that M (A M* (B) A) = M (A) B M (A) and M* (B M (A) B) = M* (B) A M* (B) for every pair A ∈ As, B ∈ Bs. Then there exist an invertible bounded linear or conjugate-linear operator T : H → H and a constant c ∈ {- 1, 1} such that M (A) = c T A T*, A ∈ As, and M* (B) = c T* B T, B ∈ Bs.

KW - Elementary operator of length one

KW - Self-adjoint operator

KW - Standard -operator algebra

UR - http://www.scopus.com/inward/record.url?scp=33750820692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750820692&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2006.04.039

DO - 10.1016/j.jmaa.2006.04.039

M3 - Article

AN - SCOPUS:33750820692

VL - 327

SP - 302

EP - 309

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -