prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$

Question

I have to prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$ (adjoint)

I know that $<Tv,u> = <\lambda v,u> = \bar\lambda<v,u>=<v,\bar\lambda u>$

and that $<Tv,u>=<v,T^*u>$

but does this imply that there is a $u$ such that $T^*u=\bar\lambda u ?$

I believe something is wrong here. Any help?

Answer 1

If $Tv=\lambda v$, with $v\ne 0$, then, for all $u\in V$, $$\langle T^*u-\bar\lambda u,v\rangle=\langle T^*u,v\rangle-\langle\bar\lambda u,v\rangle=\langle u,Tv\rangle-\lambda\langle u,v\rangle=0 $$ This means that the image of $T^*-\bar\lambda I$ is contained in the orthogonal complement of $v$, so $T^*-\bar\lambda I$ is not surjective. What can you say, now?

Of course, the assumption is that the space $T$ operates on is finite dimensional, because the assertion is false for infinite dimensional spaces.