Some properties of essential spectra of a positive operator, II

Abstract

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ ew(T), in particular, the equality σ ewT) = ∪ 0 ≤K ∞ {K} (E)} σ(T + K) , where {K}(E) is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ ef(T), then r(T) notin σ(T + a|T{-1}|) for every a ≠ 0, where T -1 is a residue of the resolvent R(., T) at r(T). The new conditions for which r(T) σ rm ef}(T) implies r(T) σew(T) = ∪0 ≤ K ∞ double strok K sign}(E) ≤ T σ (T - K), are derived. The question when the relation σew(T) ⊂ σel(T) holds, where σel(T) = ∪ 0 ≤ Q ≤ T /Q ≤ K K}(E)}σ (T - Q) is Lozanovsky's essential spectrum, will be considered. Lozanovsky's order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol'sky's theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≠ 0 is of finite rank. Under the natural assumptions (one of them is r(T) \σef(T) a theorem about the Frobenius normal form is proved: there exist T-invariant bands E = Bn ⊂ B{n - 1} ⊂ B0 = {0\ such that if r(P {Di}TP{Di}) = r(T) , where Di = Bi \cap B{i - 1}d}, then an operator P{Di}TP {Di} on D i is band irreducible. © 2008 Birkhäuser Verlag Basel/Switzerland.