by mathlover
Last Updated February 11, 2019 09:20 AM

Assume there is no eigenvalues of $T$. Let $\lambda$ is an approximate eigenvalue but not eigenvalue of a self-adjoint operator $T$. That means there exist sequences $\{x_{n}\}$ unit vectors in $\mathcal{H}$ such that $\|(T-\lambda I)x_{n}\| \to 0$. Can there exist $y_{n}$ orthogonal to $x_{n}$ i.e $\langle y_{n},x_{n}\rangle=0$., such that $\|(T-\lambda I)y_{n}\| \to 0$, with $\|y_{n}\|=1$.

On $\ell^{2}$ let $Te_n=\frac 1 n e_n$ where $(e_n)$ is the usual basis. Then $(T-0I)e_n \to 0$ and $(T-0I)e_{n+1} \to 0$; of course $\langle e_n, e_{n+1} \rangle =0$ for all $n$.

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