functional analysis

Spectra of averages of unitary representations of LCA groups

arXiv:2607.11148

summary

The paper studies the spectrum of operators obtained by averaging unitary representations of locally compact Abelian groups with respect to a probability measure, establishing inclusion results and conditions for equality, and proving a weak spectral mapping theorem for such averages.

Abstract

Let $G$ be a locally compact Abelian (LCA) group with dual group $I^$ , and let $I^¼$ 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 $I^¼$ -average $V := \int_{G} U (t) d I^¼ (t)$ (defined in the strong topology of $H$ ). We prove that $I ¨ (V) \subset I^^¼ (I^)$ 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 $U (I^½) := \int_{G} U (t) d I^½ (t)$ , where $I^½$ is any bounded complex measure on $G$ . For a unitary representation of $Z$ , defined by the powers of a unitary operator $U$ , we prove that $I ¨ (V) = I^¼ (I ¨ (U))$ . For a unitary representation of $R$ , given as $U (t) = e^{i tB}$ ( $t \in R$ ), we show that $I ¨ (V) = I^¼ (I ¨ (B))$ .

Topics & keywords

#unitary representations #locally compact abelian groups #spectral theory #averaging operators #spectral mapping theoremLCA groupsprobability measurespectral measureStone theoremFourier transformoperator spectrum