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)
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 $V$.