Title: Some  Progress on the Problem of Invariant Subspaces

Abstract:  An bounded linear operator T on Hilbert spaces  is called intransitive if it leaves invariant spaces other than 0 or the whole space; otherwise it is transitive. The  invariant  subspaces  is a big open problem In operator theory which is raised by von Neuuman. In 1970,P. R. Halmos listed ten problems in Hilbert spaces. In 1974, Problem 7 was solved by C. Apostol and D. Voiculescu .In 1976, Problem 8 was solved by D. Voiculescu . In 1997, Problem 6 was solved by G. Pisier . For Problem 5, C.Cowen and Hua Sun. The third problem is the following: Problem 3 If an intransitive operator has an inverse, is its inverse also intransitive? Thus this problem is closely related to the invariant subspace problem of Hilbert spaces. Another problem asked by P. R. Halmos is: if T is such that T² has a nontrivial invariant closed subspace, must it be the case for T too? In 2007, C. Foias, I. B. Jung, E. Ko. and C. Pearcy study the relation between the Invariant subspace problem and this problem , By the way,V. I. Lomonosov and V. S. ShulmanBy  gave a survey paper .Recently, we give an affirmative answer to Problem 3 .