Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions
  We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial, p, is noncommutative plurisubharmonic if and only if it has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are finite and f_j, k_j, F are all noncommutative analytic. We also present a theory of noncommutative integration for noncommutative polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper by Greene proves that if the noncommutative complex hessian of p takes positive semidefinite values on a "noncommutative open set" then the noncommutative complex hessian takes positive semidefinite values on all matrix tuples. Thus, p has the form above. The proof in the subsequent paper draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions. 
Jeremy M. Greene, J. William Helton, and Victor Vinnikov
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