00-221 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 (38K, LaTeX 2.09, uuencoded) 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 one-parameter 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:=w-limpt (I+Q_t)^{-1}, Y_0:= strong-lim tQ_t^{-1}, X_t:= strong-lim 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$together one can consider$aV-b, 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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