Spectra of averages of unitary representations of LCA groups

Abstract

Let $G$ be a locally compact Abelian (LCA) group with dual group $Γ$, and let $μ$ be a probability measure on (the Borel sets of) $G$. Given a unitary representation $\{U(t): t \in G\}$ in a complex Hilbert space $H$, we study the spectrum of the $μ$-average $V:=\int_G U(t)dμ(t)$ (defined in the strong topology of $H$). We prove that $σ(V) \subset \overline{\hatμ(Γ)}$ and give a sufficient condition for equality. Using the spectral measure $E(\cdot)$ given by the general Stone theorem, we prove a (weak) spectral mapping theorem for the operators $U(ν):=\int_GU(t)dν(t)$, where $ν$ is any bounded complex measure on $G$. For a unitary representation of $\mathbb Z$, defined by the powers of a unitary operator $U$, we prove that $σ(V)={\widehatμ}(σ(U))$. For a unitary representation of $\mathbb R$, given as $U(t)={\rm e}^{itB}$ ($t\in\mathbb R$), we show that $σ(V)=\overline{{\widehatμ}(σ(B))}$.