Let's say we have a self-adjoint, densly defined closed linear operator acting on a separable Hilbert space [itex] H [/itex](adsbygoogle = window.adsbygoogle || []).push({});

[tex] A:D_{A}\rightarrow H [/tex]

Let [itex] \lambda [/itex] be an eigenvalue of A and let

[tex] \Delta_{A}\left(\lambda\right) = \{\left(A-\lambda \hat{1}_{H}\right)f, \ f\in D_{A}\} [/tex]

How do i prove that

[tex] D_{A}\perp \Delta_{A}(\lambda) \Leftrightarrow \bar{\Delta_{A}(\lambda)} \neq H [/tex].

Daniel.

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# A question on Hilbert space theory

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