L-Weakly Compact Friendly Operator And M-Weakly Compact Friendly Operatior On Banach Lattice

The invariant subspace problem is one of famous problem in Functional Analysis. On the L-weakly compact operator's and M-weakly compact operator's property and invariant subspace problem are denoted in this paper. Then we introduce one operator which is L-weakly compact friendly operator and M-weakly compact friendly operator.In the first part, the backgrounds and preliminary about many operators are given briefly.In the second part, we consider the L-weakly compact operator and M-weakly compact operator's properties, then we consider every L-weakly compact operator and M-weakly compact operator on Banach lattice have non-trivial closed invariant subspace. Ie if E is a Banach lattice such that either E or E'has order continuous norm, then every bounded operator that commutes with a positive L-weakly compact operator and positive M-weakly compact operator on E has a non-trivial closed invariant subspace.In the third part, we introduce two operators which is L-weakly compact friendly operator and M-weakly compact friendly operator.Then consider whether they have non-trivial closed invariant subspace. ie Let T:Eâ†'Ebe a positive operator on a AL-or AM-space. If T commutes with a positive operator R:Eâ†'E which is dominated by a L-and M-weakly compact positive operator K:Eâ†'E, then T has a non-trivial closed invariant subspace.