Let $N$ be a normal matrix.

Now I consider a perturbation of the matrix by another self-adjoint matrix $A.$

If both were self-adjoint, we'd have the nice Weyl inequalities linking the eigenvalues of the sum.

The perturbed matrix shall be called $M=N+A$ but is not assumed to have any nice structure besides the fact that we assume that $\Re(\sigma(M)) \le -\delta$ for some $\delta>0$.

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 $d(i\lambda,\sigma(N))\le \varepsilon.$

Does this give us any information about how large $\delta$ can be in terms of $\varepsilon$ or are they independent?