Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum are introduced in the article. The conditions by which implies either or are investigated, where σ_ef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series of the resolvent R(., T) of a positive operator T around of the point λ  =  r(T) are studied. The example of a positive integral operator T : L₁→ L_∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator.