Let $ T:\ell^2 (\Bbb Z_{N})\to\ell^2 (\Bbb Z_{N})$ be defined by $(T(z))(n)=z(n+1)-z(n)$

Question

Let $T$ be a linear transformation of $\ell^2 (\Bbb Z_{N})$, for some given integer $N>0$. I am being asked to find all the eigenvalues of $T$ but I don't quite know how to approach this. Can I get some help? I feel like there is too less information.

Answer 1

If $Tz = \lambda z$, then

$$ \lambda z(n) = (Tz)(n) = z(n+1) - z(n) $$

so $z(n+1) = (\lambda + 1)z(n)$ for all $n$.

Let $z$ be an eigenvector; if $z(1) = 0$, then $z(2) = 0$, implying $z(3) = 0$ and so forth. Thus $z(1) \neq 0$, and so we can assume it is $1$ (as eigenvectors are only determined up to scale). In this case, $z(n) = (1 + \lambda)^{n-1}$. Under what circumstances does a sequence defined this way belong to $\ell^2(Z_N)$?