Let $N$ be a matrix with the following property

$N$ is diagonalizable, has spectrum only on the imaginary axis and its eigenvectors $v_1,..,v_{2n}$ are such that always $\operatorname{span}\{v_{2i-1},v_{2i}\} \perp \operatorname{span}\{v_{2j-1},v_{2j}\}$ for $j \neq i.$

Now I consider a perturbation of this matrix by another negative semi-definite 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 $\delta>0$.

Moreover, we have that $\Vert A (v_{2i-1}) \Vert + \Vert A (v_{2i}) \Vert \le \varepsilon$ for some $\varepsilon,$ i.e. two of the above eigenvectors are almost in the nullspace of $A.$

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