Lp-essential spectral theory of ordinary differential operators with almost constant coefficients

Abstract

In this paper investigation is conducted of various essential spectra of minimal, maximal and intermediate ordinary differential operators in scale of Lebesque spaces $L^p(a,ınfty)$, $1\le p\le ınfty$, obtained by means of relatively small perturbations of differential operators with constant coefficients of order $n$ by differential operators of the same order, which generalizes the results [1–3]. This makes it possible to prove the new analogons of the classical Weyl theorem of invariance of essential spectrum as well as to obtain the precise formulas for calculating essential spectra of various classes of ordinary differential operators in Lebesque spaces $L^p$. In contemporary mathematical literature a few assertions are known as Weyl's theorem (see, for example, survey [4]). The classical Weyl theorem states that if $A$ and $B$ are self-adjoint and $A-B$ is compact then ${\sigma }_e(A)={\sigma }_e(B)$, where ${\sigma }_e$ is the essential spectrum of an operator. Generalization of Weyl theorem on various essential spectra for closed operators in Banach spaces and special classes of perturbations is dealt with in papers [5–7].