The Invariant Subspace Problem

The invariant subspace problem is the simple question: "Does every bounded operator T on a separable Hilbert space H over C have a non-trivial invariant subspace? " Here non-trivial subspace means a close subspac different from {0} and different form H. The problem is easy to state, however, it is still open.The answer is ' no ' in general for (separable) complex Banach spaces. P. Enflo and C. J. Read constructed some operators without non-trivial subspace respectively. But V. Lomonosov point out those operators were given by P. Enflo and C. J. Read are not conjugate of one operator. So it is still not known whether there is a bounded linear operator on reflexive Banach space which has only the trivial invariant subspace.Lomonosov's theorem: if a non-scalar bounded operator T on a Banach space commutes with a non-zero compact operator, T has a non-trivial hyper-invariant subspace (this means, a subspace which is invariant under every operator that commutes with T). This is the main affirmative result for operators on general Banach spaces. A number of extentensive and application of Lomonosov's theorem have been obtained by several mathematicians. With Lomonosov's theorem, it seems like that we can give an affirmative answer to the invariant subspace problem. Unfortunatly, after a few years, in 1980, Hadvin-Nordgren-Radjavi-Rosenthal gave an example of an operator which does not commute with any compact operator.For separable complex Hilbert space, the problem has a part affirmative answer. In 1971, T. B. Hoover proved that if A is a n-normal operator, A≠λI, then A has a non-trivial hyperinvariant subspace. Scott Brown made use of new technique to show in 1978 that every subnormal operator has a non-trivial invariant subspace. There are many other existence theorems that have been proven using a variety of techniques, for example, Beurling's theorem, Riesz's theorem and so on.The behavior of some operators with respect to some of their invariant subspaces is the essence of structure theorems, such as the Jordan canoniacl form theorem in finite dimensional space and the spectral theorem for normal operators on Hilbert space. The study of operators and operator algebra determined by some subspaces is doing while the study of the invariant sub-space problem. If T is a reflexive operator, then T has a non-trivial invariant subspace. Von Neumann's double commutant theorem can be viewed as a statement about invariant subspaces of self-adjoint algebras of operators; the study of reflexive algebras provides analogues in the setting of non-self-adjoint algebras.Questions concerning the existence of invariant subspaces for particular classes of operators have produced a wealth of interesting theorems and examples. Many mathematicians have study the property of operators which have a non-trivial subspace. The study is very interesing and significative, and may be helpful for sloving the invariant subspace problem. In this paper, we call that operators, which are similar to the restrictions on their invariant subspaces, are crystal operators. We prove that if A～A (?) A, then A is reflexive.