# Thread: Real eigenvalues of linear operator

1. ## Real eigenvalues of linear operator

Let L : D--->H be a linear operator defined on a domain D in a complex Hilbert space H. Given that L is self-adjoint, show that its eigenvalues are real.

I know Lf=af where a is the eigen value of L and f is the eigenfunction. L is self-adjoint <Lf,g>=<f,Lg>
How do i show eigenvalues are real? many thanks

2. Originally Posted by charikaar
Let L : D--->H be a linear operator defined on a domain D in a complex Hilbert space H. Given that L is self-adjoint, show that its eigenvalues are real.

I know Lf=af where a is the eigen value of L and f is the eigenfunction. L is self-adjoint <Lf,g>=<f,Lg>
How do i show eigenvalues are real? many thanks

When working with orthonormal basis you know that $L^{*}=\overline{L^t}$ , and if $\lambda$ is an eignevalue of $L$ then $\overline{\lambda}$ is an eigenvalue of $L^{*}$ , so...

Tonio

3. Also, suppose $\lambda$ is an eigenvalue for self adjoint operator L. Then $Lv= \lambda v$ for some v with norm 1.
Now $\lambda= \lambda= <\lambda v, v>= $. Since L is self adjoint, $= = = \overline{<\lambda v, v>}= \overline{\lambda}= \overline{\lambda}$.