I Parameterization of linear operators on the holomorphisms

1. Mar 20, 2017

Whiteboard_Warrior

Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear maps on the space of holomorphisms? If so, could someone provide a proof of this? If not, what are some counterexamples?

P.S. I promise I'll post more interesting questions in the future :)

2. Mar 20, 2017

bump?