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Operator means, barycenters, and fixed point equations.

Dániel Virosztek1

  • 1HUN-REN Alfréd Rényi Institute of Mathematics, Reáltanoda U. 13-15, Budapest, H-1053 Hungary.

Acta Scientiarum Mathematicarum
|January 6, 2025
PubMed
Summary
This summary is machine-generated.

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Articles linked to this work by shared authors, journal, and citation graph.

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Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples.

Acta scientiarum mathematicarum·2025
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This survey explores the intersection of algebraic and geometric approaches to defining the mean of positive operators. It highlights how these distinct mathematical viewpoints can converge to characterize operator means.

Area of Science:

  • Operator Theory
  • Functional Analysis
  • Mathematical Physics

Background:

  • Kubo-Ando's axiomatic approach to means of positive operators.
  • The algebraic nature of their foundational work.
  • The geometric perspective of defining means via distances and centers of mass on the cone of positive operators.

Purpose of the Study:

  • To survey and highlight instances where algebraic and geometric approaches to operator means coincide.
  • To bridge the gap between axiomatic and geometric characterizations of operator means.
  • To explore the interplay between different mathematical frameworks for defining operator means.

Main Methods:

  • Reviewing axiomatic definitions of operator means.
  • Analyzing geometric interpretations involving distances and fixed-point equations.
Keywords:
BarycenterFixed point equationOperator mean

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  • Identifying common ground and convergence points between algebraic and geometric strategies.

    • Main Results:

    • Demonstration of cases where geometric methods yield fixed-point equations that align with algebraic definitions.
    • Illustrations of how the geometric viewpoint naturally complements the axiomatic framework.
    • Identification of the mathematical conditions under which algebraic and geometric approaches to operator means are unified.

    Conclusions:

    • The algebraic and geometric approaches to operator means are not mutually exclusive but can be complementary.
    • Understanding the convergence of these approaches deepens the theory of operator means.
    • This survey provides a unified perspective on operator means, valuable for researchers in operator theory and related fields.