Solution
= Solution
By the <spectral radius formula>, $r(T)<1$. Choose
$$
r(T)<r<s<1.
$$
The formula gives a constant $C$ such that $\lVert T^n\rVert\leq Cr^n$ for every $n$. Since $f$ is holomorphic on the unit disc, the <Cauchy estimate> on the circle of radius $s$ gives $|a_n|\leq M_s s^{-n}$. Therefore
$$
\sum_{n\geq0}|a_n|\lVert T^n\rVert
\leq CM_s\sum_{n\geq0}(r/s)^n<\infty.
$$
Thus $\sum_{n\geq0}a_nT^n$ converges absolutely in operator norm.