Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

Huang, Sen-Zhong

doi:10.1090/S0002-9939-99-05016-9

Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered
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Proc. Amer. Math. Soc. 127 (1999), 1473-1482 Request permission

Abstract:

We establish the following result.

Theorem. Let $\alpha :G\to {\mathcal L}(X)$ be a $\sigma (X,X_*)$ integrable bounded group representation whose Arveson spectrum $\operatorname {Sp}(\alpha )$ is scattered. Then the subspace generated by all eigenvectors of the dual representation $\alpha ^*$ is $w^*$ dense in $X^*.$ Moreover, the $\sigma (X,X_*)$ 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 $\sigma (X,X_*)$ dense in $X.$ Hence, the representation $\alpha$ is almost periodic whenever it is strongly continuous.

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