Domain of double adjoint

Question

For $T$ a densely defined linear, not necessarily bounded operator on a Hilbert space $\mathscr{H}$, and $T^{**}$ the adjoint of $T$'s adjoint, I read somewhere that $\text{ran}(T)=\text{ran}(T^{**})$. Is that true? I know that $\text{ran}(T)\subset \text{ran}(T^{**})$ because of $\text{dom}(T)\subset \text{dom}(T^{**})$, which also gives $\ker(T)\subset \ker(T^{**})$, but I wouldn't know how to prove $\text{ran}(T^{**})\subset \text{ran}(T)$.

Answer 1

Your result is true if $T$ is a closed operator (i.e. the graph of $T$ is a closed subspace of $H \times H$) because we can prove that the closure of $T$ (ie: the smallest closed extension of $T$), $\overline T = T^{**}$ ($\overline T = T$ if $T$ is a closed operator). For a proof (easy only) you may see any standard book which deals with unbounded operators for eg: Reed and Simon, Functional Analysis Vol 1, Theorem VIII.1 (revised and enlarged version).