On the invariant subspaces on Banach spaces and Banach lattices

Invariant subspace problem is one of the most important problems on Banach spaces.Very few positive results are known about it. Firstly, here was interested in thisproblem. An alternative approach was presented to the characterization of invariantsubspace problem for adjoint operators by introducing the topologies which are similarto strong and weak operator topologies. It was studied for operator algebras oninvariant subspace problem following the method of Lomonosov and Branges.It was studied on closed ideals which are invariant under a collection of operators inspace L(E) of continuous linear operators from a Banach lattice E into itself. There aresome known results that guarantee the existence of a nontrivial closed invariant ideal fora quasinilpotent positive operator on an AM- space with unit or a Banach lattice whosepositive cone contains an extreme ray. One of the main results of this work is to obtainexistence of closed ideals which are invariant under a certain collection of compactpositive operators or quasinilpotent positive operators.Compressionally decomposability was investigated on Banach lattices. For an E Banachlattice which is an AM or AL- space , it is obtained that weak compact, quasinilpotentpositive operator of L(E) is compressionally decomposable.Keywords: Invariant subspace problem, closed ideal, Banach lattice, compact operator,decomposability.JURY:1. Doç. Dr. Ömer GÖK ( Supervisor ) Date: 05.06.20062. Prof. Dr. Mehmet BAYRAMOĞLU Page: 333. Yard. Doç. Dr. Filiz KANBAY