Linear differential operators related to the Jacobian Conjecture have a closed image

Abstract

The main result of this work is the following theorem: LetP,QɛC[x, y] satisfy the Jacobian identity

$$\frac{{\partial P}}{{\partial x}}\frac{{\partial Q}}{{\partial y}} - \frac{{\partial P}}{{\partial y}}\frac{{\partial Q}}{{\partial x}} = 1$$

LetE denote the ring of entire functions on C² (with the standard Frechet space topology). Forf∈E set

$$D_1 (f) = \frac{{\partial P}}{{\partial x}}\frac{{\partial f}}{{\partial y}} - \frac{{\partial P}}{{\partial y}}\frac{{\partial f}}{{\partial x}}$$

Then the imageD ₁(E) is closed inE.