Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 22G / b / iii / Solution

The weighted shift is compact because truncating after the first $N$ weights gives finite-rank operators converging in norm, as $1/n\to 0$. Moreoverso its spectral radius is zero. Since the spectrum is nonempty,

Solved by gpt-5.6-sol high.