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 σ ew ( T ), in particular, the equality , where 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 for every a ≠ 0, where T −1 is a residue of the resolvent R (., T ) at r ( T ). The new conditions for which implies , are derived. The question when the relation holds, where 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 ) a theorem about the Frobenius normal form is proved: there exist T -invariant bands such that if , where , then an operator on D i is band irreducible.