Absztrakt: | B.Sz.-Nagy and C. Foia\c{s} have given a description of invariant
subspaces of completely nonunitary contractions in terms of regular
factorizations of their characteristic function. Since it is
rather difficult to look over all regular factorizations of a
contractive analytic function even in the simplest cases {\it
S. O. Sickler} [Indiana Univ. Math. J. 24, 635-650 (1975; Zbl
0374.47002)] initiated to derive more explicit descriptions of
invariant subspaces. His result concerns $C\sb{11}$-contractions
with scalar-valued characteristic function, and has been generalized
by {\it P. Y. Wu} [J. Oper. Theory 1, 261-272 (1979; Zbl 0431.47007)]
to $C\sb{11}$-contractions with a finite matrix characteristic
function. Extending these investigations the present paper provides
a characterization of the hyperinvariant subspaces of arbitrary
$C\sb{11}$-contractions, describes the biinvariant subspaces
of $C\sb{11}$-contractions which are weakly similar to unitaries,
and gives a description of all invariant subspaces of those contractions
whose characteristic function has a scalar multiple. These results
are achieved by studying the connection of a $C\sb{11}$-contraction
to the attached canonical unitary operator. |