This is a follow up question to this post. Consider the space $L_{p}[0,\infty)$ and the operator $(Tf)(t)=f(t+a)$ for $a>0$. How could one go about finding the point spectrum of such an operator? I understand that this means finding the eigenvalues or $\lambda$ such that
$$\tag{1}(Tf)(t)=\lambda f(t)$$ is satisfied.
Taking inspiration from daw's answer we can write,
$\tag{2} \|f\|^p = \sum_{k=0}^{n}\int_{(ka,(k+1)a)} f^p dt= \lim_{n\to\infty}\sum_{k=0}^n\int_{(ka,(k+1)a)} f^p dt.$
Here is where I'm stuck, I'm dealing with a left shift operator but I don't know how to implement that as well as the assumption that $\lambda$ is an eigenvalue for $f$. I don't know specifically how I can use the left shift to rewrite
$$ \int_{(ka,(k+1)a)} f^p dt$$ in terms of $\lambda$. Any help would be appreciated.
Attempted solution:
By assumption $\lambda$ is an eigenvalue of $f$ so, $$\tag{3} \int_{(ak,(k+1)a)} f^p dt = \lambda^n \int_{(0,1)} f^p dt$$
So (2) and (3) yield, \begin{align*} \|f\|^p &= \int_{(0,1)} f^pdt \cdot \lim_{n\to\infty}\sum_{k=1}^{n}\lambda^k. \end{align*}
Since $f\in L_{p}(\mathbb{R}_{+})$ and the LHS is finite we must have that $|\lambda|<1$ for (1) to hold.
Assuming $a>0$, the point spectrum is $\{\lambda\in\Bbb C:|\lambda|<1\}$ for $1\le p<\infty$ and $\{|\lambda|\le1\}$ for $p=\infty$.
Suppose first that $p<\infty$ and $f\in L^p$. Then it's clear that $||T^nf||_p\to0$; if $Tf=\lambda f$ this implies $\lambda^n\to0$ so $|\lambda|<1$.
Conversely, suppose $|\lambda|<1$. There exists $\alpha$ with $\Re\alpha>0$ and $$\lambda=e^{-\alpha a}.$$Let $$f(t)=e^{-\alpha t};$$then $f\in L^p$ since $p<\infty$ and $Tf=e^{-\alpha a}f=\lambda f$.
We leave the modifications for $p=\infty$ to you...