I have a continuous nonlinear operator $f:X\rightarrow X$ with $X$ an infinite dimensional Banach space. $\lambda I - f$ is a homeomorphism for all $|\lambda|\geq 1$. I also have a linear operator $B$ with norm $||B||<1$, and I know that $\lambda I - Bf$ is proper for all $|\lambda|\geq 1$.
I would like to know if $I - Bf$ is necessarily a homeomorphism. I am unable show it; $f$ has the troublesome property of being differentiable everywhere except at the origin, so I cannot use results about diffeomorphisms or the Neuberger spectrum.
Thanks!