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

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?

• No, your arguments does not show the existence of such $u$.
– gerw
Dec 19, 2013 at 20:40
• Really? I think that such a $u$ exists by using the fact that the operator $T^*- \bar{\lambda}\cdot {\rm Id}$ is not surjective, and hence it has a nontrivial kernel.
– emeu
Dec 19, 2013 at 20:46
• Is your vector space finite dimensional? Dec 19, 2013 at 20:48

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.

• I don't understand: why doesn't it work in infinite dimension? We still have the fact that non-surjectivity implies non-injectivity, haven't we?
– emeu
Dec 19, 2013 at 20:59
• @emeu No; an operator can be injective and not surjective in infinite dimensional Hilbert spaces. Did you look at the counterexample? Dec 19, 2013 at 21:00
• Ok I just saw the counter-example, thanks!
– emeu
Dec 19, 2013 at 21:03

Your argument does not imply $$T^*(u)=\overline{\lambda}u$$ because you just showed $$\langle v, \overline{\lambda}u \rangle = \langle v, T^*(u) \rangle$$ for one specific $$v$$ (an eigenvector corresponding to $$\lambda$$).

In general,

$$\langle v, u \rangle = \langle v, w \rangle$$ for some $$v\in V$$ does not imply $$u=w$$. (this is clear if you take $$v=0$$).

However, $$\langle v, u \rangle = \langle v, w \rangle$$ for all $$v\in V$$ does imply $$u=w$$.