Kérchy László honlapja

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Cím:Injection of shifts into contractions.
Szerző:K\'erchy, L.
Forrás:Acta Sci. Math. 53, No.3/4, 329-338 (1989).
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) 
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