# Characterisation of the spectrum of certain unitary representations on $L^2(G)$

I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I have that the spectrum of unitary operators are subsets of the circle group $\mathbb{T}$. Can anyone point me in the direction of a reference to these theorems?

Thanks

• If it helps, I am mostly concerned with translation operators such as the left and right regular representations. – Henry Shearman May 3 '13 at 4:50

Suppose that $G$ is abelian. The Fourier transform gives an isomorphism $L^2(G) \cong L^2(\widehat{G})$, where $\widehat{G}$ is the character group of $G$. Under this isomorphism, translation by $g \in G$ goes to the multiplication operator which multiplies a function $\hat{f}(\chi)$ on $\widehat{G}$ by the function $\chi \mapsto \chi(g)$. (Here $\chi$ is an element of $\widehat{G}$.) So the spectrum of the translation operator given by $g$ equals the spectrum of the multiplication operator by $\chi(g)$, which equals the set of values $\chi(g)$ ($\chi \in \widehat{G}$).
If $G$ is compact, then the spectrum of translation by $g$ will be the closure of the union of the spectrum of $g$ on each finite-dim'l irrep. of $G$. (Assuming I haven't blundered, this follows from the Peter--Weyl theorem.)
The non-abelian, non-compact case will be harder. The first ingredient will be the Plancherel Theorem, which describes $L^2(G)$ (assuming that $G$ admits such a theorem). You could see my answer here for some more information about this case.