I'm working on Amann's Analysis exersice VII.3.8:
Prove $\exp\in C^1(\mathcal{L}(E),\mathcal{L}(E))$ and $\partial\exp(0)=id_{\mathcal{L}(E)}$ .
Where $E$ is a Banach space, and $\mathcal{L}(E)$ is the bounded linear operator Banach algebra on $E$. The $\exp$ operator is defined by $$\exp(A)=\sum_{k=1}^\infty{A^k\over k!}.$$
My problem:
I don't even know how to calculate its derivative, though I know it should be a map from $\mathcal L(E)$ to $\mathcal{L}(\mathcal{L}(E),\mathcal{L}(E))$. I guess there should be $\partial\exp(A)(X)=\exp(A)$, but I don't know why.
Can anyone help me?