Let $X$ be an Banach space and let $g:X \to X$ be a map such that:
1) $g$ is non linear and compact (i.e. if $B$ is a bounded subset of $X$ then $g(B)$ is a precompact subset of $X$)
2) the function $f:=id+g : X \to X$, is a global homeomorphism (where id is the identity map of $X$).
Is it true that $f^{-1}$ is of the same type of $f$ i.e. identity plus a compact map?
(This is certainly true if $g$ is linear)