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Cím:On a conjecture of Teodorescu and Vasyunin.
Szerző:K\'erchy, L.
Forrás:Special classes of linear operators and other topics, 11. Int. Conf. Oper. Theory, Buchar./Rom. 1986, Oper. Theory, Adv. Appl. 28, 169-172 (1988).
Nyelv:English
Absztrakt:[For the entire collection see Zbl 0633.00014.] \par Let T be a contraction on a Hilbert space H. Assume that T is of class $C\sb{10}$ (i.e., $\lim \Vert T\p nx\Vert \ne 0=\lim \Vert T\sp{*\sp n}x\Vert$ for every nonzero $x\epsilon\sp H$). Then $(x,y)\sb{\sim}=\lim (T\sp nx,T\sp ny) (x,y\epsilon\sp H)$ defines an inner product in H, and T is isometric with respect to $(\cdot,\cdot)\sb{\sim}$. Let $\tilde T$ be the continuous extension of T to the completion $\tilde H$ of $(H,(\cdot,\cdot)\sb{\sim})$. If a vector x($\ne 0)$ in H is noncyclic for the extension $\tilde T,$ then x is also noncyclic for T, and so T has an invariant subspace. The author gives a counterexample to the following conjecture of Teodorescu and Vasyunin [Linear and complex analysis. Problem book, Lecture Notes Math. 1043, 155-157 (1984; Zbl 0545.30038)]: For every $C\sb{10}$-contraction T, the extension $\tilde T$ has a noncyclic vector x belonging to H. Let f be a function in the Hardy space $H\sp{\infty}$ such that $\Vert f\Vert\sb{\infty}\le 1$. Assume that $0<m(\alpha)<1$ and ess ran ($f\vert \alpha)\ne {\bbfT}$, where $\alpha =\{\zeta \in {\bbfT}:\vert f(\zeta)\vert =1\}$ and m is the normalized Lebesgue measure on the unit circle ${\bbfT}$. Then it is shown that for every cyclic invariant subspace H of the analytic Toeplitz operator $T\sb f$, the restriction $T=T\sb f\vert H$ is $C\sb{10}$- contraction such that every x($\ne 0)$ in H is cyclic for $\tilde T$.} \RV{K.Takahashi 
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