Pseudoirreducible and pseudoprimitive bounded operators

Abstract

Given two K-positive bounded operators T,N, where K is a closed normal generating cone of a Banach space, we say that T is K- N-pseudoirreducible if the following implications holds: αu−Tu∈K, α>0, α∈ R, u∈K, Nu≠0 ⇒ u∈K d , where K d is the d-interior of K. These operators are not necessarily K-irreducible, but outside the null space of N they behave like K-irreducible operators. Thus, some spectral properties of the K-N-pseudoirreducible operators resemble those of the K-irreducible ones. Similarly, we say that T is K-N-pseudoprimitive if for every u∈ K, Nu≠0, there is an integer p= p( u)>0 such that T k u∈ K d for k⩾ p. In this paper we analyze properties of these new class of operators. In particular, we prove that K-N-pseudoprimitive operators have dominant spectral radius, and that a K-N- pseudoirreducible operator is convergent if and only if it is K-N-pseudoprimitive.