TY - JOUR

T1 - Noncommutative plurisubharmonic polynomials part I

T2 - Global assumptions

AU - Greene, Jeremy M.

AU - Helton, J. William

AU - Vinnikov, Victor

N1 - Funding Information:
* Corresponding author. E-mail addresses: j1greene@math.ucsd.edu (J.M. Greene), helton@math.ucsd.edu (J.W. Helton), vinnikov@cs.bgu.ac.il (V. Vinnikov). 1 Research supported by NSF grants DMS-0700758, DMS-0757212. 2 The material in this paper is part of the PhD thesis of Jeremy M. Greene at UCSD. 3 Research supported by NSF grants DMS-0700758, DMS-0757212, and the Ford Motor Co. 4 Partially supported by a grant from the Israel Science Foundation.

PY - 2011/12/1

Y1 - 2011/12/1

N2 - We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1,x2,...,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y). We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e.,. for all X,H ε (Rn×n)g for every n≥1. In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form. where the sums are finite and fj, kj, F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an "nc open set" then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.

AB - We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1,x2,...,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y). We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e.,. for all X,H ε (Rn×n)g for every n≥1. In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form. where the sums are finite and fj, kj, F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an "nc open set" then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.

KW - Noncommutative analytic function

KW - Noncommutative analytic maps

KW - Noncommutative plurisubharmonic polynomial

UR - http://www.scopus.com/inward/record.url?scp=80053232634&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2011.08.006

DO - 10.1016/j.jfa.2011.08.006

M3 - Article

AN - SCOPUS:80053232634

VL - 261

SP - 3390

EP - 3417

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -