2010/12/30 by Jeremy M. Greene, Greene, Jeremy M.
Mathematics · #32A99 #32H99 #46L07 #46L89 #47A56 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.CV #math.FA #math.OA #msc:32A99 #msc:32H99 #msc:46L07 #msc:46L89 #msc:47A56
paper · pdf · doi:10.48550/arxiv.1101.0111
26 pages
arxiv created 2011/01/14 · arxiv updated 2011/01/17
We say that a symmetric noncommutative polynomial in the noncommutative free variables (x1, x2, ..., xg) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if a noncommutative polynomial is noncommutative plurisubharmonic on a noncommutative open set, then the polynomial is actually noncommutative plurisubharmonic everywhere and has the form p = ∑ fjT fj + ∑ kj kjT + F + FT where the sums are finite and fj, kj, F are all noncommutative analytic. In the paper by Greene, Helton, and Vinnikov, it is shown that if p is noncommutative plurisubharmonic everywhere, then p has the form above. In other words, the paper by Greene, Helton, and Vinnikov makes a global assumption while the current paper makes a local assumption, but both reach the same conclusion. This paper uses a Gram-like matrix representation of noncommutative polynomials. A careful analysis of this Gram matrix plus the main theorem in the paper by Greene, Helton, and Vinnikov ultimately force the form in the equation above.