I am wondering how to show that:

if $A$ is a non-negative operator, then $A$ is self-adjoint.

Def. 1. $A$ is non-negative if $\langle Ax,x \rangle \geq 0$ for $\forall x\in H$, where $H$ is a Hilbert space.

Def. 2. $A$ is self-adjoint if $A = A^*$.

  • $\begingroup$ The key here is that $H$ must be a complex inner product space. For a real inner product space these properties are not equivalent; see the discussion under Does non-symmetric positive definite matrix have positive eigenvalues? $\endgroup$ – hardmath Jun 10 '17 at 16:01
  • $\begingroup$ Thanks @hardmath; how can prove for $H$ real inner product space? $\endgroup$ – Amin Jun 10 '17 at 16:14
  • $\begingroup$ It is not true for real inner product spaces, as the example given in answer to the previous Question I linked shows. I can point out other Questions with such examples, if it will help. $\endgroup$ – hardmath Jun 10 '17 at 16:16
  • $\begingroup$ @hardmath can you provide the links for that. $\endgroup$ – Amin Jun 10 '17 at 16:19
  • 1
    $\begingroup$ Have a look at the example in the Accepted Answer for Non-symmetric positive-definite matrices. $\endgroup$ – hardmath Jun 10 '17 at 16:22

For any linear $A:H \rightarrow H $, we have

$\langle A^{*}x, x\rangle = \langle x, Ax\rangle =\overline {\langle Ax, x \rangle},$

But, for $A $ non-negative, then $\langle Ax, x\rangle$ is real, so

$\langle Ax, x\rangle = \overline {\langle Ax, x \rangle}$

i.e. $\langle Ax, x\rangle =\langle A^{*}x, x\rangle $

$\implies \langle (A-A^{*})x, x\rangle = 0, \forall x \in H$.


Claim: If $\langle Tx, x\rangle = 0 \: \forall x $ in a complex Hilbert space, then $T=0$.


Pick any $u, v \in H $, and let $T:H \rightarrow H $ such that $\langle Tx, x\rangle = 0 \: \forall x \in H $.


$ 0 = \langle T(u+v), u+v\rangle = \langle Tu, v\rangle + \langle Tv, u\rangle$

$ \implies - \langle Tu,v\rangle = \langle Tv,u\rangle $,


$0 = \langle T(u+iv), u+iv\rangle = i\langle Tv, u\rangle - i\langle Tu, v\rangle$

$ \implies \langle Tu,v\rangle = \langle Tv,u\rangle$.

Then $\langle Tu, v\rangle = - \langle Tu, v\rangle $

i.e. $\langle Tu, v\rangle = 0 \: \forall u,v \in H. $

$\implies T = 0. $


Hence, $A=A^{*} $.


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