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Completeness of Eigenvectors of Group Representations of Operators Whose Arveson Spectrum is Scattered

Proceedings of the American Mathematical Society / v.127 no.5. 1999, pp.1473-1482

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Abstract : We establish the following result. Theorem. Let $\alpha $ : G $\rightarrow {\cal L}$ (X) be a $\sigma $ (X, X $_{\ast}$ ) integrable bounded group representation whose Arveson spectrum Sp( $\alpha $ ) is scattered. Then the subspace generated by all eigenvectors of the dual representation $\alpha ^{\ast}$ is w $^{\ast}$ dense in X $^{\ast}$ . Moreover, the $\sigma $ (X, X $_{\ast}$ ) closed subalgebra W $_{\alpha}$ generated by the operators $\alpha _{t}$ (t $\in $ G) is semisimple. If, in addition, X does not contain any copy of c $_{0}$ , then the subspace spanned by all eigenvectors of $\alpha $ is $\sigma $ (X, X $_{\ast}$ ) dense in X. Hence, the representation $\alpha $ is almost periodic whenever it is strongly continuous.

Keyword : Primary 47A67, 47A10 . Spectrum of group representation . Almost periodicity

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