Kérchy László honlapja

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Cím:On the reducing essential spectra of contractions.
Szerző:K\'erchy, L\'aszl\'o
Forrás:Acta Sci. Math. 57, No.1-4, 175-198 (1993).
Nyelv:English
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) 
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