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

Research output: Article

9 Citations (Scopus)

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 languageEnglish
Pages (from-to)3058-3065
Number of pages8
JournalLinear Algebra and Its Applications
Volume430
Issue number11-12
DOIs
Publication statusPublished - jún. 1 2009

Fingerprint

Harmonic mean
Hilbert spaces
Positive Operator
Mathematical operators
Positive Linear Operators
Bounded Linear Operator
Invertible
Linear Operator
Automorphisms
Hilbert space
Form

ASJC Scopus subject areas

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

Cite this

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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.",
keywords = "Arithmetic mean, Automorphisms, Harmonic mean, Parallel sum, Positive operators",
author = "L. Moln{\'a}r",
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AU - Molnár, L.

PY - 2009/6/1

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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

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

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ER -