Let $X$ denote the space $L^{1}(\mathbb{R})$ of all (equivalence classes of) Lebesgue integrable functions $f:\mathbb{R} \to \mathbb{C}$ with the norm $||f||_{1} = \int_{\mathbb{R}}|f(t)|dt$. Let $T \in B(X)$ be defined by
$$ (Tf)(t)=f(t+1), \, \, \, \, f \in L^{1}(\mathbb{R}), t \in \mathbb{R} $$
Find the point spectrum, the approximate spectrum and the spectrum of T.
I have managed to show that $T$ is a unitary operator which means the spectrum must lie in the unit circle $S^{1}$. I am also fairly sure the point spectrum must be empty as it would required an eigenvector which is not integrable over $\mathbb{R}$.
I am struggling to see what to do to find the approximate point spectrum or the spectrum in general.
Thank you for any help!