 00221 Sergej A. Chorosavin ( sergius@pve.vsu.ru )
 A Nonlinear Approximation of Operator Equation $V^{*}QV=Q$ :
Nonspectral Decomposition of Nonnormal Operator
and Theory of Stability
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May 11, 00

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Abstract. $V$ denote arbitrary bounded bijection on Hilbert space $H$.
We try to describe the sets of $V$stable vectors, i.e. the set of elements $x$
of $H$ such that the sequence $\V^N x\ (N=1,2,...)$ is bounded (we also
consider some other analogous sets). We do it in terms of oneparameter
operator equation $ Q_t=V^*(Q_t+tI)(I+tQ_t)^{1}V , 0\leq Q$,
($t$ is real valued parameter $0\leq t \leq 1$,$Q$ is operator to be found $).
Definition: for $t \to +0 $ denote
$R_0:=wlimpt (I+Q_t)^{1}, Y_0:= stronglim tQ_t^{1}, X_t:= stronglim tQ_t $
In the case of the normal $V$ it is noted that the operators $X_0,Y_0,R_0$
define (in essential) the spectral subspaces of $V$
(with $V$ together one can consider $aVb, b/a \not\in spectrum V$).
In this article we will show that the similar situation holds for the
arbitrary bounded bijection $V$.
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