A note on a problem of Abramovich, Aliprantis and Burkinshaw
POSITIVITY, vol.15, no.3, pp.473-480, 2011 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 15 Issue: 3
- Publication Date: 2011
- Doi Number: 10.1007/s11117-010-0096-2
- Journal Name: POSITIVITY
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.473-480
- Istanbul Kültür University Affiliated: Yes
Abstract
Let B and T be two positive operators on a Banach lattice such that B is compact-friendly and T is locally quasi-nilpotent. Introducing the concept of positive quasi-similarity, we prove that T has a non-trivial closed invariant subspace provided B is positively quasi-similar to T. This gives an affirmative answer to a problem of Abramovich, Aliprantis and Burkinshaw with the commutativity condition replaced by the positive quasi-similarity of the corresponding operators. The notion of strong compact-friendliness is also introduced and relevant facts about it are discussed.