Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known that this implies that $d(\sigma(K),\lambda) \le \varepsilon.$
However, I am wondering whether this implies also something about $x.$
For example, it seems plausible that $x$ cannot be orthogonal to the direct sum of eigenspaces of $K$ with eigenvalues that are close to $\lambda.$
In other words, are there any non-trivial restrictions on $x$ coming from the spectral decomposition of $K$?