# Range of self-adjoint operator

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!

• This is only true if $z\ne0$ – 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.