ABSTRACT

INVARIANT SUBSPACES OF POSITIVE OPERATORS ON BANACH LATTICES

Çaglar, Mert
M.Sc., Department of Mathematics
Supervisor: Assoc. Prof. Dr. Zafer Ercan

May 2001, 58 pages

In this thesis, we study the invariant subspaces of positive operators on Banach lattices. First, we give the existence theorem -and results related to it- of an invariant ideal for a positive operator $T$ on a Banach lattice such that there exists a positive operator $S$ satisfying $ST\leq TS$, $S$ is quasinilpotent at some $x_{0}$, and $S$ dominates a non-zero AM-compact operator. Then we introduce the concept of weak quasinilpotence and prove as an extension of a result of Abramovich, Aliprantis and Burkinshaw that each continuous linear operator with modulus, on a Hausdorff locally convex solid Riesz space $E$ with compact order intervals, which is weak quasinilpotent at some $0<x\in E$, has a non-trivial closed invariant ideal.

Keywords: Riesz Space, Banach Lattice, Positive Operator, Quasinilpotence, Weak Quasinilpotence.