In a Hilbert space if $T$ is a self-adjoint operator and $\lambda\in \mathbb{R}$ is an eigenvalue of $T$ which is in a spectral gap $(r,s)$, then $spectrum(T)\subseteq(-\infty,r]\cup[s,\infty)$ and if $T$ is invertible we can say that $||T^{-1}||\leq \frac{1}{dist(\lambda,spectrum(T))}$ ..
Now if we still work on $T$ but with $\lambda\in \mathbb{{C|R}}$ then $T-\lambda$ is a normal operator and hence we still have the relation with the distance above..
my question is how the spectrum of T with complex $\lambda$ looks like in this case and do we still have the fact that $spectrum(T)\subseteq(-\infty,r]\cup[s,\infty)$ ??