| Absztrakt: | For contractions T with $\Vert T\Vert <1$, injection of shifts
S into T, i.e., the relation $XS=TX$ with an injection X, was
investigated by {\it B. Sz.-Nagy} and {\it C. Foia\c{s}} [Proc.
M. R. I. Oberwolfach, Birkh\"auser, 29-37 (1974; Zbl 0302.47011)]
and {\it C. Apostol}, {\it H. Bercovici}, {\it C. Foia\c{s}}
and {\it C. Pearcy} [Mich. Math. J. 29, 243-255 (1982; Zbl 0493.47013)].
This paper treats injection of shifts into contractions T with
$\Vert T\Vert =1$. Let T be a completely non- unitary contraction
on a Hilbert space ${\cal H}$ and $\mu\sb{*,T}=\sup \{rank(I-\theta
(\zeta)\theta (\zeta)*):\vert \zeta \vert =1\}$, where $\theta$
is the characteristic function of T. The number $\mu\sb{*,T}$
coincides with the multiplicity of the *-residual part of the
minimal unitary dilation of T. The author proves that if $\mu\sb{*,T}<\infty$,
then there exists a set $\Phi$ of injections such that $\cup
\{ran X;\quad X\in \Phi \}={\cal H}$ and $XS=TX$ for each $X\in
\Phi$, where S is a unilateral shift of multiplicity $\mu\sb{*,T}$.}
\RV{K.Takahashi (Sapporo) |