Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 4 / a / Solution

By the spectral radius formula, $r(T)<1$. ChooseThe formula gives a constant $C$ such that $\Vert T^n\Vert \le C{r}^n$ for every $n$. Since $f$ is holomorphic on the unit disc, the Cauchy estimate on the circle of radius $s$ gives $|a_n|\le M_s{s}^{-n}$. ThereforeThus ${\sum }_{n\ge 0}{a}_n{T}^n$ converges absolutely in operator norm.