I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.

More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\epsilon}$, i.e. $\sum_{n \ge 1}n||T_n||< \infty$, and $T'(z)=\sum_{n \ge 1}nT_nz^{n-1}$ is $\epsilon$-Holder on the closed unit disk. It can be proved easily that $||T_n||=O(n^{-1-\epsilon})$.

Is there a theory to study the asymptotic expansion of $T_n$, namely, $n^{1+\epsilon}T_n$ converges to some operator?