Some Properties of Essential Spectra of a Positive Operator, Ii

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σew(T ), in particular, the equality σew(T ) = ⋂ 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 σef(T ), then r(T ) 6∈ σ(T +a|T−1|) for every a 6= 0, where T−1 is a residue of the resolvent R(., T ) at r(T ). The new conditions for which r(T ) 6∈ σef(T ) implies r(T ) 6∈ σ− ew(T ) = ⋂ 0≤K∈K(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 ) 6∈ σef(T )) a theorem about the Frobenius normal form is proved: there exist T -invariant bands E = Bn ⊇ Bn−1 ⊇ . . . ⊇ B0 = {0} such that if r(PDiTPDi) = r(T ), where Di = Bi ∩Bd i−1, then an operator PDiTPDi on Di is band irreducible. Mathematical Subject Classification. 47B65, 47A55, 47A11, 47A10, 47A53