Plurisubharmonic noncommutative rational functions

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

doi:10.1016/j.jmaa.2020.124421

Plurisubharmonic noncommutative rational functions

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

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

A noncommutative (nc) function in x₁,…,x_g,x₁^⁎,…,x_g^⁎ 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 x_j^⁎) 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 language	English
Article number	124421
Journal	Journal of Mathematical Analysis and Applications
Volume	492
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2020.124421
State	Published - 1 Dec 2020
Externally published	Yes

Keywords

Convex function
Free analysis
Noncommutative rational function
Plurisubharmonic function
Realization

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2020.124421

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title = "Plurisubharmonic noncommutative rational functions",

abstract = "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.",

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AU - Dym, Harry

AU - Helton, J. William

AU - Klep, Igor

AU - McCullough, Scott

AU - Volčič, Jurij

PY - 2020/12/1

Y1 - 2020/12/1

N2 - 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.

AB - 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.

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KW - Free analysis

KW - Noncommutative rational function

KW - Plurisubharmonic function

KW - Realization

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Plurisubharmonic noncommutative rational functions

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Keywords

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