A Spectral Mapping Theorem for Representations of Compact Groups

Abstract

Let $U$ be a strongly continuous bounded representation of a locally compact group $G$ on a Banach space $E$. For a bounded regular Borel measure $\mu$ on $G$, we denote by $U(\mu )$ the operator $U(\mu )=\int U(t)d\mu (t)$. If $G$ is abelian, it is known that

holds if the continuous singular part of it is zero (where $\sigma (U(\mu ))$ denotes the spectrum of the operator $U(\mu )$, sp $(U)$ the Arveson-spectrum of $U$ and $\hat{\mu }$ the Fourier–Stieltjes transformation of $\mu$.)

In the present article a corresponding spectral mapping theorem is proved for compact (non-abelian) groups and absolutely continuous measures. Moreover, it is shown that - in contrary to the abelian case - the spectral mapping theorem fails for purely discontinuous measures.