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

Author: Sen-Zhong Huang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1473-1482
MSC (1991): Primary 47A67, 47A10
Published electronically: January 29, 1999
MathSciNet review: 1621945
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Abstract | References | Similar Articles | Additional Information

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

Keywords: Spectrum of group representation, almost periodicity
Received by editor(s): September 1, 1997
Published electronically: January 29, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society