A Nonlinear Approximation of Operator Equation $V^{\ast }QV=Q$ : 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 $\Vert V^{N}x\Vert (N=1,2,...)$ is bounded (we also consider some other analogous sets). We do it in terms of one-parameter operator equation $Q_t=V^{\ast }(Q_t+tI)(I+t{Q}_t{)}^{-1}V,0\le Q$, ($t$ is real valued parameter $0\le t\le 1$,$Q$ 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 $Inthecaseofthenormal$V$itisnotedthattheoperators$X_0,Y_0,R_0$define(inessential)thespectralsubspacesof$V$(with$V$togetheronecanconsider$aV-b, b/a \not\in spectrum V$).Inthisarticlewewillshowthatthesimilarsituationholdsforthearbitraryboundedbijection$V$.

In the Section 3, Observation 1 corrected. Latex 2.09