Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 7A / c / Solution

The eigenvalue equation is $u_{i+1}=\lambda u_i$, soThis is a nonzero square-summable vector exactly when $c\ne 0$ and $|\lambda |<1$. Hence every point of the open unit disc is an eigenvalue, with the displayed one-dimensional eigenspace.