Symmetric cone monotone functions and symmetric cone convex functions

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Abstract

Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r' (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).

Original languageEnglish
Pages (from-to)499-512
Number of pages14
JournalJournal of Nonlinear and Convex Analysis
Volume17
Issue number3
Publication statusPublished - 2016 Jan 1

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Keywords

  • Euclidean Jordan algebra
  • Lowner operator
  • Matrix-monotone
  • SC-convex
  • SC-monotone
  • Symmetric cone

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Control and Optimization
  • Applied Mathematics

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