I recently posted a question here about adjoint maps. I now want to take this further and show that - with the assumptions as in the original - the minimum polynomials of $T$ and $T^{*}$ are the same. Further to this, what can be said of the eigenvalues of $T$ and $T^{*}$?
This is false over $\mathbb{C}$. You need to specify that you're working over $\mathbb{R}$. – Qiaochu Yuan Jan 12 '13 at 1:33