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Cím:Invariant subspaces of $C\sb{1\sp .}$-contractions with non-reductive unitary extensions.
Szerző:K\'erchy, L.
Forrás:Bull. Lond. Math. Soc. 19, 161-166 (1987).
Nyelv:English
Absztrakt:Let T be a contraction acting on the Hilbert space H such that $\lim\sb{n}\Vert T\sp nh\Vert \ne 0$, for every $0\ne h\in H$. Then $[x,y]=\lim\sb{n}<T\sp nx,T\sp ny>$ (x,y$\in H)$ defines a new scalar product on H, and T acts as an isometry on the new scalar product space (H,[.,.]). Let $\tilde H$ denote the completion of (H,[.,.]), and let $\tilde T$ be the continuous extension of T from (H,[.,.]) to $\tilde H.$ It is proved that if the isometry $\tilde T$ is non-reductive, i.e. if $\tilde T$ contains a unilateral shift, then T possesses a non-trivial invariant subspace. This result partially answers the question of {\it R. Teodorescu} and (INVALID INPUT)V. Vasyunin posed in [Linear and complex analysis. Problem book (1984; Zbl 0545.30038)]. \par Futhermore, relying on a theorem of {\it H. Bercovici} and {\it K. Takahashi} [J. Lond. Math. Soc., II. Ser. 32, 149-156 (1985; Zbl 0536.47009)], it is shown that if in addition to the previous assumptions even $\lim\sb{n}\Vert T\sp{*n}h\Vert \ne 0$ holds, for every $0\ne h\in H$, then T is reflexive. This theorem extends former results of {\it P. Y. Wu} [Proc. Am. Math. Soc. 77, 68-72 (1979; Zbl 0417.47001)] and the author. 
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