Normal operator close in norm to projection, what about the spectrum? Let $T$ be a normal operator on a Hilbert space and suppose $T$ is close in norm to a projection $P$. Can I say that the spectrum of $T$ is contained in small balls around $0$ and $1$?

First of all $\rho(T)=||T||$ because $T$ is normal ($\rho$ is the spectral radius). So by the triangle inequality $\rho(T)\leq1+c$, given $||T-P||\leq c$.
Now the formula in the comment 
$$(T-\lambda I)^{-1}=((T-P)(P-\lambda I)^{-1}+I)^{-1}(P-\lambda I)^{-1}$$
makes sense, using the Von Neumann series, if $(T-P)(P-\lambda I)^{-1}$ is less than $1$ in norm.
If we let $A$ be the closed ball centered at $0$ of radius $1+c$, with two small open balls removed around $0$ and $1$, we have $\sup_A ||(P-\lambda I)^{-1}||<\infty$ by compactness. So if $c$ is small enough we have the result. The smaller the open balls, the smaller the $c$.
