The Lower Weyl Spectrum of a Positive Operator

For the lower Weyl spectrum $$sigma_{rm w}^-(T) = bigcap_{0 le K in mathcal{K}(E) le T} sigma(T - K),$$ where T is a positive operator on a Banach lattice E, the conditions for which the equality ${sigma_{rm w}^-(T) = sigma_{rm w}^-(T^*)}$ holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ? _∞ such that the spectral radius ${r(T) in sigma_{rm w}^-(T) {setminus} (sigma_{rm f}(T) cup sigma_{rm w}^-(T^*))}$ , where σ _f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T _?1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at ${r(T) notin sigma_{rm f}(T)}$ , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, ${r(T) notin sigma_{rm f}(T)}$ , can not have the trivial band ${E_n^sim}$ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from ${E_sigma^sim otimes F}$ , where F is a Banach lattice, ${E_sigma^sim}$ is a disjoint complement of the band ${E_n^sim}$ of E*.