Spectral properties of band irreducible operators

Abstract

Number of spectral properties of a band irreducible operator $T$ on a Banach lattice $E$ will be discussed. If $T$ is $\sigma$-order continuous, $r(T)$ is a pole of the resolvent $R(.,T)$, and the band $E_c^\tilde$ of all $\sigma$-order continuous functionals on $E$ is nonzero, then we prove among others that $r(T)>0$, that $T$ has an eigenvector which is a weak unit, and that the adjoint $T^*$ of $T$ has a positive order continuous eigenvector. Furthermore, we provide some criteria of primitivity for band irreducible operators in terms of limits of real sequences. Finally, we discuss the question whether the operator inequalities $0<S<T$ imply the spectral radius inequality $r(S)<r(T)$, where $T$ is a band irreducible operator on $E$.