Noncommutative plurisubharmonic polynomials part I: Global assumptions

作者：Greene Jeremy M; Helton J William^*; Vinnikov Victor

来源：Journal of Functional Analysis, 2011, 261(11): 3390-3417.

DOI：10.1016/j.jfa.2011.08.006

• Noncommutative analytic function
• Noncommutative analytic maps
• Noncommutative plurisubharmonic polynomial

We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x(1), x(2), ..., x(g)}. We define the nc complex hessian of p as the second directional derivative (replacing x(T) by y)
q(x,x(T))[h,h(T)]: = partial derivative(2)p/partial derivative s partial derivative t (x + th, y + sk) vertical bar t, s = 0 vertical bar y=x(T.) k=h(T).
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 a x a matrices for every size a; i.e.,
p= Sigma f(j)(T) f(j) + Sigma k(j) k(j)(T) + F + F(T) (0.1)
where the sums are finite and f(j), k(j), 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 sernidefinite 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.

• 出版日期2011-12-1