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

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

Answers 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 $n$.

Kavi Rama Murthy
Kavi Rama Murthy
February 11, 2019 09:12 AM

Related Questions

Updated July 01, 2017 04:20 AM

Updated November 21, 2017 03:20 AM

Updated December 11, 2017 06:20 AM

Updated March 04, 2019 08:20 AM

Updated March 19, 2019 18:20 PM