On The Invariant Subspace Problem

Master thesis" On the invariant subspace problem" is the organic combinarive result of the study on the theory of Banach space and the operator theory of functional analysis.There are three chapters in the paper.Chapter 1 mainly introduces some basic results about the paper,including some conceptions and notations needed in the following chapters.The main works of this paper are presented in chapter 2 and chapter 3.In chapter 2,using the free ultrafilter,we show that the contracting operators on the reflexive Banach space have nontrivial invariant subspaces under certain condition.In chapter 3,we define the subinvariant subspace of the operators on the Hilbert space and prove that every operator on Hilbert space has nontrivial subinvariant subspace.