Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\Vert (N-i\lambda)u \Vert \le \varepsilon$
for some $\lambda \in \mathbb R.$
Since $N$ is normal this implies that $d(i\lambda,\sigma(N))\le \varepsilon.$
Moreover, assume that $\Re(\sigma(M)) \le -\delta$ for some $\delta>0.$
If we assume additionally that $Au=0.$ Does this give us any information about how large $\delta$ can be in terms of $\varepsilon$ or are they independent?