Let $N=iT$ be a normal matrix where $T$ is self adjoint, i.e. $N$ is skew self-adjoint.

Now I consider a perturbation of this matrix by another negative semi-definite self-adjoint matrix $A$ with $0 \in \sigma(A) \subset(-\infty,0].$

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

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

Moreover, we assume the following property:

There is a normalized eigenvector $v$ of $N$ with $\Vert A v \Vert \le \varepsilon$ for some $\varepsilon,$ i.e. one of the eigenvectors of $N$ is (almost) in the nullspace of $A.$

Equivalently you can work with the assumption, that there is an approximate eigenvector $v$ for $N$ such that

$$\Vert (N-\lambda)v \Vert \le \varepsilon \ \text{ and } Av=0.$$

The question:

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