A Note On A Problem Of Abramovich, Aliprantis And Burkinshaw - Çağlar, Mert; Mısırlıoğlu, Remzi Tunç (Springer, Van Godewijckstraat 30, 3311 Gz Dordrecht, Netherlands, 2011)
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