Invariant Subspace Theorems for Positive Operators

We establish new invariant subspace theorems for positive operators on Banach lattices. Here are three sample results. • If a quasinilpotent positive operator S dominates a non-zero compact operator K (i.e., |Kx| ≤ S |x| for each x), then every positive operator that commutes with S, in particular S itself, has a non-trivial closed invariant ideal. • If a positive kernel operator commutes with a quasinilpotent positive operator, then both operators have a common non-trivial closed invariant subspace. • Every quasinilpotent positive Dunford-Pettis operator has a non-trivial closed invariant subspace.