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).