discrete translationally invariant operator and its eigenfunctions

Question

Let $\ell^2 := \{f:\mathbb{Z} \rightarrow \mathbb{C} : \sum_{k\in\mathbb{Z}}|f(k)|^2 < \infty\}$.

Let $L$ be a linear, bounded, translationally invariant operator $L:\ell^2\rightarrow \ell^2$, i.e. $$LT=TL,$$ where $T:\ell^2\rightarrow \ell^2$ is the translation operator defined by $(Tf)(x) := f(x+1)$ for every $f$.

What are the eigenfunctions of $T$? If $T$ is defined on $\{f:\mathbb{Z} \rightarrow \mathbb{C} \}$ then the eigenfunctions would be $e_k:\mathbb{Z}:\rightarrow \mathbb{C}$ with $e_k(x) = e^{ikx}$. What happens when it is restricted to $\ell^2$?

And another question is, when are the eigenfunctions of $L$ identical to the ones of $T$? To proof that, I can use the spectral theorem. Thus I need $L$ to be normal. But can it happen that $L$ is not normal?

(This question is correlates to linear translationally invariant operator represented as a stencil. , but I'm pretty confused right now...)

Answer 1

To answer the first part of your question, the spectral values of $T:\ell_p\to\ell_p$ are, in fact, $\{z\in \Bbb C:|z|=1\}$, because you can show that $T-zI$ is continuously invertible if and only if $|z|\ne 1$. However, if $p\ne\infty$, then these spectral values are not eigenvalues (i.e. there are no corresponding eigenvectors).