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 |