Prove that if a function f $\in \mathcal{L}(V,V)$ is nilpotent then its adjoint $f^{*}$ is also nilpotent and they have the same integer such that $f^m=0$ and $(f^{*})^m=0$.
Should I begin this problem with the scalar product and with the help of a orthogonal basis? And I also only know that the matrix representation equation $[f]_{B,B}^H=[f^*]_{B,B}$, where the notation H is hermitian transpose and B the orthogonal basis of V. Could you please give me some advices?