# 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 $\displaystyle L^{*}=\overline{L^t}$ , and if $\displaystyle \lambda$ is an eignevalue of $\displaystyle L$ then $\displaystyle \overline{\lambda}$ is an eigenvalue of $\displaystyle L^{*}$ , so...

Tonio

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