Is a Self-adjoint Operator Always Real?

[daudaudaudau]

Hello.

I have a linear operator, $$L$$, and its adjoint $$L^a$$. $$L$$ is self-adjoint, so $$L=L^a$$. I'm being told that the following is true:

$$\langle f,Lh\rangle=\langle Lf,h\rangle$$.

But what if the scalar product is not the symmetric product? What if

$$\langle f,h\rangle=\langle h,f\rangle^*$$

where $$^*$$ is complex conjugation ? Then my first equation tells me that

$$\langle f,Lf\rangle=\langle Lf,f\rangle$$.

and the second one says that

$$\langle f,Lf\rangle=\langle Lf,f\rangle^*$$.

But which is true?

 

[gel]

But which is true?

Both, <f,Lf> is real.

 

[daudaudaudau]

But how do you know?

 

[gel]

But how do you know?

You just showed that it equals its complex conjugate, <f,Lf> = <Lf,f>* = <f,Lf>*. So it is real.
Quite a standard result (eg, self adjoint operators represent real valued observables in quantum mechanics).