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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 2, Issue 2, 1983, pp. 127–133**

**DOI: 10.4171/ZAA/54**

Published online: 1983-04-30

A Spectral Mapping Theorem for Representations of Compact Groups

Wolfgang Arendt^{[1]}and Claudio D'Antoni

^{[2]}(1) Universität Ulm, Germany

(2) Università dell’Aquila, Italy

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 $$\sigma (U(\mu)) = \hat {\mu} (\mathrm {sp} U))^–$$ 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.

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Arendt Wolfgang, D'Antoni Claudio: A Spectral Mapping Theorem for Representations of Compact Groups. *Z. Anal. Anwend.* 2 (1983), 127-133. doi: 10.4171/ZAA/54