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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 $K$?

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  • $\begingroup$ 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, $\sum_{j : |\lambda_j - \lambda| > k \varepsilon} |\alpha_j|^2 \leqslant 1/k^2$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 5, 2019 at 10:40

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