Let $T:\mathrm{dom}(T) \subseteq H \to H$ be a densely defined, self-adjoint operator on a Hilbert spaces $$. In general the range of $T$ is not guaranteed to be closed. What tools are available to check if the range is closed? More precisely, what are a list of equivalent formulations of closed range, or conditions that imply closed range?



Characterising closed range self-adjoint operators
This is a complete, but rather abstract characterisation.



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> $T$ has closed range if and only if there is $H\_0\subseteq H$ closed and a bounded self-adjoint injective map $R:H\_0\rightarrow H\_0$ with $D(T) = \{ \xi+R(\eta) : \xi\in H\_0^\perp, \eta\in H\_0 \}$ and $T(\xi+R(\eta)) = \eta$.
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If this holds, then $T$ is well-defined as $R$ injective, and the image of $T$ is $H\_0$ which is closed. The graph of $T$ is $G(T) = \{ (\xi+R\eta, \eta):\xi\in H\_0^\perp, \eta\in H\_0\}$ which can be shown to be closed. Then $(\alpha,\beta)\in G(T^\*)$ if and only if $((-\beta,\alpha)|(\xi+R\eta,\eta))=0$ for $\xi\in H\_0^\perp, \eta\in H\_0$. This is equivalent to $(\beta|\xi)=(\alpha-R\beta|\eta)$ for all $\eta\in H\_0, \xi\in H\_0^\perp$, which is equivalent to $\alpha-R\beta\in H\_0^\perp, \beta\in H\_0^{\perp\perp}=H\_0$, that is, $(\alpha,\beta)\in G(T)$. So $T$ is self-adjoint.


Conversely, let $T$ be self-adjoint with closed range $H\_0$. Then $H\_0 = (\ker T)^\perp$ so $\xi\in H\_0^\perp \implies \xi\in D(T), T\xi=0$. Define $S:D(S)\rightarrow H\_0$ by $D(S)=D(T)\cap H\_0$ and $S(\xi) = T(\xi)$. Then one can show that $S$ is closed, injective, self-adjoint, and onto. Then $R=S^{-1}:H\_0\rightarrow H\_0$ is closed and everywhere defined (so bounded), self-adjoint, and injective (and so automatically has dense-range). $T$ has the form stated above, using $R$.


