Let $E$ be a Banach space. Let $A \in L(E)$, the space of linear operators from $E$. Show that the linear operator $\varphi: L(E) \to L(E)$ with $\varphi (T) = T + AT$ is an isomorphism if $\|A\| < 1$.
So the idea here is to use the Neumann series but I can't really figure out how to apply it here. Any help?
You have to note that $\varphi (T)=(I+A)T $. As $\|A\|<1$, the element $I+A $ ($=I-(-A) $) is invertible, with inverse $$ B=\sum_{k=0}^\infty (-1)^kA^k. $$ Then $\psi (T)=BT $ is the inverse of $\varphi $.