  The Adams operators on a Hopf algebra $H$ are the convolution powers of the
identity map of $H$. They are also called Hopf powers or Sweedler powers. It is
a natural family of operators on $H$ that contains the antipode. We study the
linear properties of the Adams operators when $H=\bigoplus_{m\in \mathbb{N}}
H_m$ is connected graded. The main result is that for any of such $H$, there
exist a PBW type homogeneous basis and a natural partial order on it such that
the restrictions $\Psi_n|_{H_m}$ of the Adams operators are simultaneously
upper triangularizable with respect to this ordered basis. Moreover, the
diagonal coefficients are determined in terms of $n$ and a combinatorial number
assigned to the basis elements. As an immediate consequence, we obtain a
complete description of the characteristic polynomial of $\Psi_n|_{H_m}$, both
on eigenvalues and their multiplicities, when $H$ is locally finite and the
base field is of characteristic zero. It recovers the main result of the paper
[2] by Aguiar and Lauve, where the approach is different to ours.
