Absztrakt: | The author brings together the notions of $C\sb{11}$-contractions
and reducing essential spectrum, for bounded operators on an
infinite- dimensional Hilbert-space $H$. First, a $C\sb{11}$-contraction
is a contraction, $A$, quasisimilar to a unitary operator $U$
in $H$, i.e., there are injective operators $X$, $Y$ with dense
image such that $$XA= UX,\ AY= YU.$$ This class of operators
deserves interest because, among other things, their spectra
can be fully characterized as the family of closed subsets, $\sigma$,
of $\overline D, D= \{z\in \bbfC\mid \vert z\vert< 1\}$, with
the property that every connected component $\sigma$ intersects
$\partial D$ in a set of positive Lebesgue measure.\par
On the
other hand, $\lambda\in \bbfC$ belongs to the reducing essential
spectrum of a bounded operator $A$ iff there is a nonzero orthogonal
projection, $P$, in $H$ such that both $(A- \lambda)P$ and $(A\sp*-
\overline \lambda)P$ are compact operators; denote this set by
$R\sb e A$. Clearly, $R\sb e A\subset \text{spec}\sb e A$.\par
The
main result (Theorem 1) of the paper asserts that $R\sb e A$
has no special structure whatsoever for absolutely continuous
$C\sb{11}$- contractions (i.e. contractions quasisimilar to an
absolutely continuous unitary): every compact subset of $\overline
D$ can arise.\par
The ideas used in the proof of this result
are also used to elaborate on the incompatibility of the $H\sp
\infty$-functional calculus developed by Foias and Sz.-Nagy with
the Calkin algebra which was pointed out by Esterle and Zarouf.}
\RV{J.Br\"uning (Augsburg) |