# Perturbation theory compact operator

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 $$K$$?

• If $\alpha_j$ are the coefficients in the eigenvector expansion of $x$, then $\sum |\alpha_j|^2 = 1$ and $\sum |\alpha_j|^2 |\lambda_j - \lambda|^2 \leqslant \varepsilon^2$. That's all one can say, and of course this implies that, for example, $\sum_{j : |\lambda_j - \lambda| > k \varepsilon} |\alpha_j|^2 \leqslant 1/k^2$. – Mateusz Kwa?nicki Mar 5 at 10:40