On uniqueness of approximate eigen sequence of linear operator on Hilbert spaces

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

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 $$\|y_{n}\|=1$$.

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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 $$n$$.