I don't understand a part of this proof, hope you can help me sort it out.
Let $T$ be a linear operator on inner product space $V$ for which the characteristic polynomial splits. Then $T^*$ has at least one eigenvector.
Proof. Because $T$ splits, it has an eigenvector $v$ with eigenvalue $\lambda$ $\Rightarrow$ $(T-\lambda I)v=0$
$\Rightarrow\langle (T-\lambda I)v, x \rangle = 0$ for all $x \in V$
$\Rightarrow\langle v, (T-\lambda I)^* x \rangle = 0$
$\Rightarrow\langle v, (T^*- \overline{\lambda}I) x \rangle = 0$
$\Rightarrow v \notin R(T^*- \overline{\lambda}I)$
$\Rightarrow R(T^*- \overline{\lambda}I) \ne V$
Therefore, $(T^*- \overline{\lambda}I)$ is not invertible (why???)
Hence, there exists an engenvector of $T^*$ (with eigenvalue $\lambda$) (why???).
I don't understand the last two conclusion.
- Why $(T^*- \overline{\lambda}I) \ne V$ implies $(T^*- \overline{\lambda}I)$ is not invertible?
- Why $(T^*- \overline{\lambda}I)$ is not invertible implies an eigenvector of $T^*$?
Thanks for any help!
Best Answer
First of all, you seem to have written the final step incorrectly. We should state that $R(T^* - \bar \lambda I) \neq V$; this is not the same as $(T^* - \bar \lambda I) \neq V$, which doesn't make much sense.
Note that a linear operator is invertible if and only if it is both one-to-one (injective) and onto (surjective). Because $(T - \bar \lambda I)$ is not surjective, it cannot be invertible.
To put that another way: the statement $v \notin R(T - \bar \lambda I)$ is the same as saying that the equation $(T - \bar \lambda I) x = v$ has no solution (for $x$). However, if $(T - \bar \lambda I)$ had an inverse, then we could use the inverse to solve the equation and get $x = (T - \bar \lambda I)^{-1}v$.
To your second question, if $(T - \bar \lambda I)$ is not invertible, then $\bar \lambda $ must be an eigenvalue of $T^*$. One way to see that this is the case is as follows: if $(T^* - \bar \lambda I)$ is not invertible, then the equation $(T^* - \bar \lambda I)x = 0$ has a solution $x \neq 0$ (i.e. a "non-trivial" solution). Expanding the left side of this equation leads to $$ T^*x - \bar \lambda Ix = 0 \implies T^*x = \bar \lambda x. $$ So, $x$ is an eigenvector of $T^*$ associated with $\bar \lambda$.