### Abstract

Let H be a complex Hilbert space. The symbol A ! B stands for the harmonic mean of the positive bounded linear operators A, B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.

Original language | English |
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Pages (from-to) | 3058-3065 |

Number of pages | 8 |

Journal | Linear Algebra and Its Applications |

Volume | 430 |

Issue number | 11-12 |

DOIs | |

Publication status | Published - Jun 1 2009 |

### Fingerprint

### Keywords

- Arithmetic mean
- Automorphisms
- Harmonic mean
- Parallel sum
- Positive operators

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Numerical Analysis

### Cite this

**Maps preserving the harmonic mean or the parallel sum of positive operators.** / Molnár, L.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 430, no. 11-12, pp. 3058-3065. https://doi.org/10.1016/j.laa.2009.01.022

}

TY - JOUR

T1 - Maps preserving the harmonic mean or the parallel sum of positive operators

AU - Molnár, L.

PY - 2009/6/1

Y1 - 2009/6/1

N2 - Let H be a complex Hilbert space. The symbol A ! B stands for the harmonic mean of the positive bounded linear operators A, B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.

AB - Let H be a complex Hilbert space. The symbol A ! B stands for the harmonic mean of the positive bounded linear operators A, B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.

KW - Arithmetic mean

KW - Automorphisms

KW - Harmonic mean

KW - Parallel sum

KW - Positive operators

UR - http://www.scopus.com/inward/record.url?scp=64649096460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64649096460&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2009.01.022

DO - 10.1016/j.laa.2009.01.022

M3 - Article

AN - SCOPUS:64649096460

VL - 430

SP - 3058

EP - 3065

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 11-12

ER -