Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered
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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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Additional Information
- Sen-Zhong Huang
- Affiliation: Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany
- Address at time of publication: Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, 18055 Rostock, Germany
- Email: huang@sun.math.uni-rostock.de
- Received by editor(s): September 1, 1997
- Published electronically: January 29, 1999
- Communicated by: David R. Larson
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1473-1482
- MSC (1991): Primary 47A67, 47A10
- DOI: https://doi.org/10.1090/S0002-9939-99-05016-9
- MathSciNet review: 1621945