The Lower Weyl Spectrum of a Positive Operator

Abstract

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 l∞ such that the spectral radius \({r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}\) , where σf(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*.