Absztrakt: | Based on the Sz.-Nagy-Foia\c{s} functional model for completely
non- unitary (c.n.u.) contractions on a separable Hilbert space,
this paper gives a systematic study of the residual and *-residual
parts of such contractions, and proves the reflexivity of certain
$C\sb{11}$ contractions. \par
To be more precise, recall that
the residual part R on space K of a c.n.u. contraction T on H
is the unitary part in the Wold decomposition of the minimal
isometric dilation of T; the *-residual part $R\sb*$ on $K\sb*$
is similarly defined in terms of $T\sp*$. The paper starts by
giving a representation for $R\sb*$ in the functional model of
T, and then considers a canonical operator X intertwining T and
$R\sb*$ and Y intertwining R and T. Here X (resp. Y) is the restriction
to H (resp. K) of the orthogonal projection onto $K\sb*$ (resp.
H). The author studies various properties of X and Y such as
when they are injective or have dense range, and relates these
to properties of T. Samples: X is injective if and only if T
can be mapped injectively into some unitary operator; Y has dense
range if and only if some unitary operator can be mapped, with
dense range, into T. Then the author defines the class of $quasi$-C${}\sb{11}$
contractions, shows by an example that they properly contain
$C\sb{11}$ contractions, and concludes the paper by proving two
reflexivity results, one for $quasi$-C${}\sb{11}$ contractions
and one for $C\sb{11}$ contractions, under certain extra conditions.}
\RV{Wu Pei Yuan |