Non-Commutative Polynomial Solutions to Partial Differential Equations

作者：Nelson Christopher S^*

来源：Integral Equations and Operator Theory, 2012, 74(4): 527-585.

DOI：10.1007/s00020-012-1998-0

This paper extends the notion of a differential equation to the space of polynomials p in non-commutative variables x = (x (1), . . . , x (g) ). The directional derivative D[p, x (i) , h] of p in x (i) is a polynomial in x and a non-commuting variable h. When all variables commute, D[p, x (i) , h] is equal to ha,p/a,x (i) . This paper classifies all non-commutative polynomial solutions to a special class of partial differential equations, including the non-commutative extension of Laplace's equation. Of interest also are non-commutative subharmonic polynomials. A non-commutative polynomial is subharmonic if its non-commutative Laplacian takes positive-semidefinite matrix values whenever matrices X (1), . . . , X (g) , H are substituted for the variables x (1), . . . , x (g) , h. This paper shows that a homogeneous subharmonic polynomial which is bounded below is a harmonic polynomial-that is, a non-commutative solution to Laplace's equation-plus a sum of squares of harmonic polynomials.

• 出版日期2012-12