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Cím:Approximation and quasisimilarity.
Szerző:K\'erchy, L.
Forrás:Acta Sci. Math. 48, 227-242 (1985).
Nyelv:English
Absztrakt:For an arbitrary complex Hilbert space, B(H) denotes the Banach algebra of all bounded linear operators on H. For any $T\in B(H)$, Alg T denotes the weakly closed subalgebra of B(H) generated by T and the identity I and let $\{$ $T\}$ ' be the commutant of T. A subspace ${\frak M}$ of H is called to be cyclic for T if $\bigvee\sb{n\ge 0}T\sp n{\frak M}=H$; ${\frak M}$ is a minimal cyclic subspace for T if it contains no proper subspace which is also cyclic for T. The number disc T is defined as the supremum of the dimensions of all finite dimensional minimal cyclic subspaces for T. For a contraction T (i.e., $\Vert T\Vert \le 1)$ on H, T is of class $C\sb{11}$ if $$ \lim\sb{n\to \infty}\Vert T\sp nh\Vert \ne 0\ne \lim\sb{n\to \infty}\Vert T\sp{*n}h\Vert$$ for every $h\in H$ and let $C\sb 1$ denote the class of contractions whose unitary part is absolutely continuous with respect to the Lebesgue measure m on the unit circle ${\bbfT}$. It is known that for any operator $T\in C\sb 1$, there exists a unique Borel subset $\alpha$ of ${\bbfT}$ such that $m(\alpha)>0$ and T is quasisimilar to $M\sb{\alpha}$, where $M\sb{\alpha}$ denotes the operator of multiplication by $\phi(\zeta)\equiv \zeta$ on $L\sp 2(\alpha).$ \par The author investigates in this paper whether the approximating property Alg T$=\{T\}'$ and the number disc T of $C\sb{11}$-contraction T are invariant under the quasiaffine transform. \par About these questions for the $C\sb 1$-contraction, the results of {\it H. Bercovici} and {\it L. K\'erchy} [Acta Sci. Math. 45, 67-74 (1983; Zbl 0526.47003)] and {\it P. Y. Wu} [J. Oper. Theory 1, 261-272 (1979; Zbl 0431.47007)] are known under some additional conditions.} \RV{T.Yoshino 
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