Spectra of averages of unitary representations of LCA groups

Abstract

Let $G$ be a locally compact Abelian (LCA) group with dual group $\Gamma$, and let $\mu$ 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 $\mu$-average $V:=\int_G U(t)d\mu(t)$ (defined in the strong topology of $H$). We prove that $\sigma(V) \subset \overline{\hat\mu(\Gamma)}$ 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 $U(\nu):=\int_GU(t)d\nu(t)$, where $\nu$ 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 $\sigma(V)={\widehat{\mu}}(\sigma(U))$. For a unitary representation of $\mathbb R$, given as $U(t)={\rm e}^{itB}$ ($t\in\mathbb R$), we show that $\sigma(V)=\overline{{\widehat{\mu}}(\sigma(B))}$.