Holomorphic self-maps of the loop space of ${\mathbb{P}}^n$

The loop space $L{\mathbb{P}}^n$ of the complex projective space ${\mathbb{P}}^n$ consisting of all $C^k$ or Sobolev $W^{k,p}$ maps $S^1\to {\mathbb{P}}^n$ is an infinite dimensional complex manifold. We identify a class of holomorphic self-maps of $L{\mathbb{P}}^n$, including all automorphisms.