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

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- Solution a

Solution

By the spectral radius formula,

r (T) < 1

. Choose

r (T) < r < s < 1.

The formula gives a constant

C

such that

∥ T^{n} ∥ \leq 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} ∣ \leq M_{s} s^{- n}

. Therefore

\sum_{n \geq 0} ∣ a_{n} ∣ ∥ T^{n} ∥ \leq C M_{s} \sum_{n \geq 0} (r / s)^{n} < \infty.

Thus

\sum_{n \geq 0} a_{n} T^{n}

converges absolutely in operator norm.

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