The Adams operators Ψn on a Hopf algebra H are the convolution powers
of the identity of H. We study the Adams operators when H is graded
connected. They are also called Hopf powers or Sweedler powers. The main result
is a complete description of the characteristic polynomial (both eigenvalues
and their multiplicities) for the action of the operator Ψn on each
homogeneous component of H. The eigenvalues are powers of n. The
multiplicities are independent of n, and in fact only depend on the dimension
sequence of H. These results apply in particular to the antipode of H (the
case n=−1). We obtain closed forms for the generating function of the
sequence of traces of the Adams operators. In the case of the antipode, the
generating function bears a particularly simple relationship to the one for the
dimension sequence. In case H is cofree, we give an alternative description for
the characteristic polynomial and the trace of the antipode in terms of certain
palindromic words. We discuss parallel results that hold for Hopf monoids in
species and q-Hopf algebras.Comment: 36 pages; two appendice