Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$.

Question (?) :Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \mathcal{O}_D$ which is not a (possibly infinite order) differential operator. I.e. not of the form:$$\phi=\Sigma_{n=0}^{\infty} b_n(z) \partial^n$$

Where $\partial =\frac{d}{d z}$ and $b_n \in \mathcal{O}_D$ .

EDIT: Suppose I require that $\phi$ be continuous w.r.t. to the natural frechet topology on $\mathcal{O}_D$ coming from uniform convergence on compact subsets, does the answer change?

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