eigenvalue question

[ee7klt]

hi,
if an operator O has the property that O^{4}f(x)=f(x), what are the eigenvalues of O? any hints on how to go about this?

[Galileo]

Since O^4 is the identity, it has all eigenvalues one.
If O has an eigenvector f with eigenvalue \lambda, what can you tell about O^2