# Plurisubharmonic Noncommutative Rational Functions

## TL;DR

This paper characterizes noncommutative rational functions that are plurisubharmonic, showing they are compositions of convex and analytic functions, with a constructive proof and a practical criterion based on minimal realization.

## Contribution

It provides a complete characterization of plush nc rational functions as compositions of convex and analytic functions, with a constructive proof and a computable criterion.

## Key findings

- Nc rational functions are plush iff they are compositions of convex and analytic functions.
- A constructive proof method is developed for the characterization.
- A simple, computable condition for plushness based on minimal realization is provided.

## Abstract

A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ 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.