Eigenvalues of a linear operator $A$, knowing all the traces of $A^n$.

Question

If I know the trace of all the $n$-th powers of a linear operator $A$ in an analytic form $f(n)= {\rm Tr} A^n$, is there a way to compute all the eigenvalues of the operator $A$? In particular, I am interested in the case for which $A$ is an adjoint operator on an infinite-dimensional Hilbert space.