A Nonlinear Approximation of Operator Equation $V^{*}QV=Q$ : Nonspectral Decomposition of Nonnormal Operator and Theory of Stability

Abstract

$V$ denotes 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 $V$.