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Cím:Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes.
Szerző:K\'erchy, L\'aszl\'o
Forrás:Acta Sci. Math. 61, No.1-4, 443-476 (1995).
Absztrakt:Let $T\in {\cal L} ({\cal H})$ be the absolutely continuous contraction such that the spectral multiplicity function of its unitary asymptote $T^{(a)}$ is at least $n$ $(1\leq n\leq \aleph_0)$ almost everywhere on a subset $\gamma$ (with positive measure) of the unit circle ${\bold T}$. Let ${\cal G}_n$ be an $n$-dimensional Hilbert space and let $J_{n, \gamma}$ denote the canonical embedding of the Hilbert space $H^2 ({\cal G}_n)$ into $\chi_\gamma L^2 ({\cal G}_n)$. It is shown that, given any positive $\varepsilon$, there exists a factorization $J_{n, \gamma}= ZY$ such that the mappings $Y\in {\cal L} (H^2 ({\cal G}_n), {\cal H})$ and $Z\in {\cal L} ({\cal H}, \chi_\gamma L^2 ({\cal G}_n))$ intertwine $T$ with the operators of multiplication by the identical function on the corresponding spaces, and the product: $|Y||Z|\leq \sqrt{2}+ \varepsilon$. As a consequence, we obtain that the unilateral shift $S_n$ of multiplicity $n$ can be completely injected into $T$ and that $T$ has the property $({\bold A}_{n, \aleph_0} (\gamma))$ introduced in the theory of dual algebras. It follows furthermore that in the special case $\gamma= {\bold T}$ the contraction $T$ has an invariant subspace ${\cal H}'$ such that the restriction $T\mid {\cal H}'$ is similar to $S_n$. 
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