Essential Spectra of Elliptic Partial Differential Equations

Let A be a closed, densely defined operator in a Banach space X. There are several definitions of the "essential" spectrum of A (cf. [ l ] , [2]). According to Wolf [3], [4] it is the complement in the complex plane of the $-set of A. The $-set $A of A is the set of points X for which (a) a(A — X), the dimension of the null space of A — X, is finite (b) R(A —X), the range of A —X, is closed (c) p(A •—X), the codimension of R(^4 — X), is finite. We denote the essential spectrum according to this definition by dew{A). The set crem(A)f as defined in [ l ] , [2] is obtained by adding to <rew(A) those points X for which a(A —X) 9^^{A —X). It is the largest subset of <r(A) which remains invariant under compact perturbations. Finally, to obtain the set aeb(A), which is the essential spectrum according to Browder [5], we add to aem{A) those points of cr(A) which are not isolated. Interest in the sets aeW(A), <rem(A), aeb(A) is centered about the fact that they remain invariant under certain perturbations of A. In particular one has