Regular operator mappings and multivariate geometric means

Frank Hansen

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)

Abstract

We introduce the notion of regular operator mappings of several variables generalising the notion of spectral function. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives. As a main application of the theory we consider geometric means of k operator variables extending the geometric mean of k commuting operators and the geometric mean of two arbitrary positive definite matrices. We propose different types of updating conditions that seems natural in many applications and prove that each of these conditions, together with a few other natural axioms, uniquely defines the geometric mean for any number of operator variables. The means defined in this way are given by explicit formulas and are computationally tractable.

Original languageEnglish
Pages (from-to)123-138
Number of pages16
JournalLinear Algebra and Its Applications
Volume461
DOIs
Publication statusPublished - 2014 Nov 15

Keywords

  • Multivariate geometric mean
  • Perspective
  • Regular operator mapping

ASJC Scopus subject areas

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

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