Kernels of linear maps: A generalization of Duistermaat and Van der Kallens theorem

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 $\operatorname{Ker} L$ is a Mathieu-Zhao space where $L$ is the linear map \begin{align*} L\colon {\bf C}[X_1,\ldots,X_n,X_1^{-1},\ldots,X_n^{-1}] \to C,\ f \mapsto f_0\end{align*}. In this paper, we generalize this result (for $n = 1$) to all non-trivial linear maps $L\colon C[X,X^{-1}] \to C$ such that $\{X^n \mid |n|\geq N\} \subset \operatorname{Ker} L$ for some $N \geq 1$.