JournalszaaVol. 2, No. 2pp. 127–133

A Spectral Mapping Theorem for Representations of Compact Groups

  • Wolfgang Arendt

    Universität Ulm, Germany
  • Claudio D'Antoni

    Università dell’Aquila, Italy
A Spectral Mapping Theorem for Representations of Compact Groups cover
Download PDF

Abstract

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

σ(U(μ))=μ^(spU))\sigma (U(\mu)) = \hat {\mu} (\mathrm {sp} U))^–

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

Cite this article

Wolfgang Arendt, Claudio D'Antoni, A Spectral Mapping Theorem for Representations of Compact Groups. Z. Anal. Anwend. 2 (1983), no. 2, pp. 127–133

DOI 10.4171/ZAA/54