Operator means, barycenters, and fixed point equations

Virosztek, Dániel (2024) Operator means, barycenters, and fixed point equations. ACTA SCIENTIARUM MATHEMATICARUM (SZEGED), 90 (3). pp. 391-408. ISSN 0001-6969

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Abstract

The seminal work of Kubo and Ando [17] provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavor. On the other hand, it is highly natural to take the geometric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.

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