If we let $f: V \rightarrow V$ be a normal linear operator on a complex inner product space $V$ such that $f^4 = f^3$, can we use the spectral theorem to prove $f$ is self-adjoint and $f^2 = f$?
I worked with the matrix representation of $f$, made the change of basis to diagonal form, $D$, and noticed that $D^4 - D^3 = 0$ implies that $D$ is either the identity or the zero transformation - I think this means that the eigenvalues are real, but cannot figure out how to prove that this is a self-adjoint linear operator. Any hints given would be appreciated.