| 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 |