Abstract. A characterization of the spectral maximal subspaces of translation operators in $L^p(G)$ spaces, where $G$ is locally compact abelian group and $1\le p\le\infty $, is given in terms of Fourier transforms. We apply this to the case of the group of integers for the values $1\le p< \infty $ and to the case of the space $c_0({\msbm Z})$. We show how the Riemann--Lebesgue localization principle for Fourier series and our results together imply the localization principle for conjugate series.

AMS Subject Classification (1991): 47B40, 47A11, 42A20

Received May 26, 1994. (Registered under 5631/2009.)