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Let $T^{*}$ be the adjoint of a densely defined, symmetric operator $T$. Suppose that the operator $T^{*}$ is a symmetric operator. Why does it follow that $T$ is essentially self-adjoint?

Thanks in advance!

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  • $\begingroup$ Are you assuming that $T$ is symmetric and densely defined as well? $\endgroup$
    – s.harp
    Commented Mar 18, 2020 at 12:42
  • $\begingroup$ Yes, I correct that now. $\endgroup$
    – user525192
    Commented Mar 18, 2020 at 12:48

1 Answer 1

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Here is a proof using the definitions.

First check that if $T$ is symmetric and densely defined, that then $D(T^*)\supseteq D(T)$ and $T^*\lvert_{D(T)}= T$. For if $x \in D(T)$ then: $$|\langle x, Ty\rangle| = |\langle Tx,y\rangle| ≤ \|Tx\| \ \|y\|$$ for all $y\in D(T)$ and $x\in D(T^*)$ follows. Further $T^*(x)$ is defined via: $$\langle T^* x, y\rangle := \langle x, Ty\rangle = \langle Tx,y\rangle$$ holding for all $y\in D(T)$ (which is dense). It follows that $T^*x=Tx$ for all $x\in D(T)$.

Now if $T^*$ is also symmetric this very same step gives you $D(T^{**})\supseteq D(T^*)$ and $T^{**}\lvert_{D(T^*)}=T^*$. The only thing we are still interested in proving is that $D(T^*)\supseteq D(T^{**})$. So let $x\in D(T^{**})$. This means that $$|\langle x,T^*y\rangle| ≤ C_x\|y\|$$ for all $y\in D(T^*)$. In particular since $D(T^*)\supseteq D(T)$ it holds for all $y\in D(T)$ and you get the desired $x\in D(T^{*})$.

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  • $\begingroup$ What definition of essential-self-adjointness are you using? $\endgroup$
    – user525192
    Commented Mar 18, 2020 at 13:48
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    $\begingroup$ I had some of the things mixed up: The general statement is that if $T$ is symmetric then $T^{**}$ is the closure of $T$. This is always true and a calculation can be found in the appendix of these notes for example. What is done here shows that $T^*$ is self-adjoint, hence $T^{**}=T^*$ is self-adjoint giving that the closure of $T$ is self-adjoint. $\endgroup$
    – s.harp
    Commented Mar 18, 2020 at 14:13
  • $\begingroup$ Thank you very much! $\endgroup$
    – user525192
    Commented Mar 18, 2020 at 14:14

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