J. Aust. Math. Soc.  78 (2005), 423-428
The spectral mapping property for p-multiplier operators on compact abelian groups

Werner J. Ricker
  Math.-Geogr. Fakultät
  Katholische Universität Eichstätt-Ingolstadt
  D-85072 Eichstätt

Let $G$ be a compact abelian group and $1 < p < \infty $. It is known that the spectrum $\sigma (T_\psi)$, of a Fourier $p$-multiplier operator $T_\psi $ acting in $L^p(G)$, may fail to coincide with its natural spectrum $\overline{\psi (\Gamma)}$ if $p \neq 2$; here $\Gamma$ is the dual group to $G$ and the bar denotes closure in $\mathbb{C}$. Criteria are presented, based on geometric, topological and/or algebraic properties of the compact set $\sigma (T_\psi)$, which are sufficient to ensure that the equality $\sigma (T_\psi) = \overline{\psi (\Gamma)} $ holds.
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