# 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 window.___gcfg = {lang: 'ko'}; (function() { var po = document.createElement('script'); po.type = 'text/javascript'; po.async = true; po.src = 'https://apis.google.com/js/platform.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(po, s); })();

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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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