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Cím:Isometric asymptotes of power bounded operators.
Szerző:K\'erchy, L\'aszl\'o
Forrás:Indiana Univ. Math. J. 38, No.1, 173-188 (1989).
Nyelv:English
Absztrakt:Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. An operator $T\in B(H)$ is called power bounded if the powers of T form a bounded sequence $\{T\sp n\}\sp{\infty}\sb{n=1}$ in B(H). Let L denote a Banach limit, that is, a positive linear functional on the sequence space $\ell\sp{\infty}({\bbfN})$ with the properties $L(1,1,...)=1$ and $L(c\sb 1,c\sb 2,c\sb 3,...)=L(c\sb 2,c\sb 3,...)$ and define $[x,y]=L(\{<T\sp nx,T\sp ny>\}\sb n),$ where T is power bounded and $x,y\in H$. Then the factor space $H/H\sb 0$, where $H\sb 0=\{x\in H:\quad [x,y]=0\},$ endowed with $[x+H\sb 0,y+H\sb 0]:=[x,y]$ will be an inner product space. Let $H\sb+\sp{(a)}$ denote the resulting Hilbert space obtained by completion. Since L is a Banach limit, $[Tx,Ty]=[x,y]$ for x,y in H, and so $T\sb 0: x+H\sb 0\to Tx+H\sb 0$ is an isometry on $H/H\sb 0$. The continuous extension of $T\sb 0$ to $H\sb+\sp{(a)}$ is called the isometric asymptote of the power bounded operator T. The minimal unitary extension of $T\sb+\sp{(a)}$ is called the unitary asymptote of T. \par In this paper isometries are invoked to investigate power bounded operators. These isometries are attached to operators in an asymptotic way and do not degenerate if the powers of the operators considered do not converge to zero in the strong operator topology. After giving the basic definitions and theorems in the first section, in Section 2 the author describes how the spectra of power bounded operators relate to the spectra of their asymptotes, while in Section 3 the connection between the hyperinvariant subspace lattices is treated.} \RV{M.Z.Nashed 
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