Kernels of linear maps

The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as Ker L is a Mathieu-Zhao space where L is the linear map $L:C[X_1,\dots ,X_n,X_1^{-1},\dots ,X_n^{-1}]\to C,f\mapsto f_0$. In this paper, we generalize this result (for $n=1$) to all non-trivial linear maps $L:C[X,X^{-1}]\to C$ such that $\left\{X^n\right|\left|n\right|\ge N\}\subset \mathrm{Ker}L$ for some $N\ge 1$.