A Spectral Mapping Theorem for Representations of Compact Groups

  • Wolfgang Arendt

    Universität Ulm, Germany
  • Claudio D'Antoni

    Università dell’Aquila, Italy

Abstract

Let be a strongly continuous bounded representation of a locally compact group on a Banach space . For a bounded regular Borel measure on , we denote by the operator . If is abelian, it is known that

holds if the continuous singular part of it is zero (where denotes the spectrum of the operator , sp the Arveson-spectrum of and the Fourier–Stieltjes transformation of .)

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