Let $T$ be a self adjoint operator in a Hilbert space $H$. Let $I$ be the identity operator on $H$ and $z\in \mathbb{R}$. Why does it hold that the range of $$T-izI$$ is $H$? Thanks in advance!

  • $\begingroup$ This is only true if $z\ne0$ $\endgroup$ – daw Jul 19 at 7:33

This is because $\sigma(T) \subseteq \mathbb{R}$, see here.

Now $iz \notin \mathbb{R}$ so $T-izI$ is invertible. In particular, $T-izI$ is surjective.

  • $\begingroup$ Thank you very much! $\endgroup$ – ShaqAttack1337 Jul 18 at 13:28

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