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?



Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?
Probably [this paper](https://www.jstage.jst.go.jp/article/kyushumfs/32/2/32_2_301/_article) answers your question negatively. 


In particular it shows that if $X$ is an open subset of $\mathbb{C}^n$ and the sheaf $\mathcal{O}$ of holomorphic functions on $X$ is given the topology of compact convergence then every continous endomorphism of $\mathcal{O}$ is given by a convergent $\mathcal{O}$-linear sum of operators $\partial\_1^{\alpha\_{1}}\cdots \partial\_n^{\alpha\_n}$ with $\alpha\_i\in \mathbb{N}$.


