Some Invariant Subspace Theorems

In this paper, we research the invariant subspace problem and obtain some invariant subspace theorems.First of all, we show that every Lower-triangular positive operators on lp(1 ≤p < ∞) space has non-trivial closed invariant subspaces, and partly solve the open problem arisen from the famous mathematical journal 5Journal of Functional Analysis6 in 1993:/Whether or not every positive operator on l1(or lpwith 1 ≤ p < ∞)has a non-trivial closed invariant subspace?0The second, we improve Riesz’s theorem about invariant subspaces, and so that Riesz Space Decomposition Theorem and Jordan Space Decomposition Theorem can be obtained as corollaries. Hence some literatures can be simplified. For example,we can modify Chapter 2 of the book5 Introduction to operator theory and invariant subspaces6 published by North-Holld, and omit Chapter 1 of this book(the result of Chapter 1 can be obtain as corollaries of Chapter 2).