Busch–Gudder metric on the Cone of Positive Semidefinite Operators and Its Isometries

  • Lajos Molnár


In this paper we introduce a new distance measure called Busch–Gudder metric on the cone of all positive semidefinite operators acting on a complex Hilbert space. It is defined as the sup-distance between the so-called strength functions corresponding to positive semidefinite operators. We investigate the properties of that metric, among others its relation to the metric induced by the operator norm. We show that in spite of many dissimilarities between the topological features of those two metrics, their isometry groups still coincide.


Positive cone Positive operators Strength functions Busch–Gudder metric Operator norm Isometries 

Mathematics Subject Classification

Primary 47B49 47B65 Secondary 54G99 



The author is grateful to the referee for his/her comments which have helped to improve the presentation of the paper.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Functional Analysis Research Group Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary

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