Let ${\displaystyle G}$  and ${\displaystyle H}$  be Hilbert spaces, and let ${\displaystyle T:\operatorname {dom} (T)\subseteq G\to H}$  be an unbounded operator from ${\displaystyle G}$  into ${\displaystyle H.}$  Suppose that ${\displaystyle T}$  is a closed operator and that ${\displaystyle T}$  is densely defined, that is, ${\displaystyle \operatorname {dom} (T)}$  is dense in ${\displaystyle G.}$  Let ${\displaystyle T^{*}:\operatorname {dom} \left(T^{*}\right)\subseteq H\to G}$  denote the adjoint of ${\displaystyle T.}$  Then ${\displaystyle T^{*}T}$  is also densely defined, and it is self-adjoint. That is,