  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 $N \geq 1$.
