## Publication: A Note On A Problem Of Abramovich, Aliprantis And Burkinshaw

2011-09
##### Authors
Çağlar, Mert
Mısırlıoğlu, Remzi Tunç
##### Publisher
Springer, Van Godewijckstraat 30, 3311 Gz Dordrecht, Netherlands
##### 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.

##### Keywords
invariant subspace , positive operator , compact-friendly , locally quasi-nilpotent , positive quasi-similarity , strongly compact-friendly , operators , değişmeyen alt uzay , pozitif operatör , kompakt dostu , lokal yarı-nilpotent , pozitif yarı-benzerlik , güçlü kompakt dostu , operatörler