Plurisubharmonic noncommutative rational functions

Harry Dym, J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A noncommutative (nc) function in x1,…,xg,x1,…,xg is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no xj) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization.

Original languageEnglish
Article number124421
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Dec 2020
Externally publishedYes


  • Convex function
  • Free analysis
  • Noncommutative rational function
  • Plurisubharmonic function
  • Realization

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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