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Cím:On the spectra of $C\sb{11}$-contractions.
Szerző:Bercovici, H.; K\'erchy, L.
Forrás:Proc. Am. Math. Soc. 95, 412-418 (1985).
Nyelv:English
Absztrakt:Assume T is a completely nonunitary contraction on Hilbert space H, of class $C\sb{11}$ (i.e., neither $T\sp nh\to 0$ nor $T\sp{*\sp n}h\to 0$ for nonzero $h\in H$, as $n\to \infty).$ \par As a slight generalization of a known fact one first deduces that for every nonempty "clopen" part $\sigma$ ' of $\sigma$ (T) (i.e., with both $\sigma$ ' and $\sigma$ (T)$\setminus \sigma '$ closed) we have $m(\sigma '\cap \sigma (R\sb T))>0$, where $R\sb T$ is the unitary part of the minimal isometric dilation of T, and m denotes Lebesgue measure on the unit circle ${\bbfT}.$ \par The main result of the paper is the following (more than a) converse of this fact. Assume $\sigma$ is a closed subset of the closed unit disc and $\Sigma\sb 0$ is a Borel subset of $\sigma\cap {\bbfT}$ such that every nonempty clopen part of $\sigma$ meets the closed support of the restriction of m to $\Sigma\sb 0$ in a set of positive measure. Then one can find a cyclic, completely nonunitary contraction T of class $C\sb{11}$ such that $\sigma (T)=\sigma$ and $R\sb T$ is unitarily equivalent to the operator of multiplication by the identical function on the space $L\sp 2(m\vert \Sigma\sb 0)$.} \RV{B.Sz\"okefalvi-Nagy 
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