Transitive Algebra Problem

An operator algebra on Hilbert space is called transitive algebra, if it is weakly closed ,contains the identity operator and has no non-trivial subspace. "Is every transitive alge-bra equal to B(H)" is called the Transitive Algebra Problem. It was raised by Kadison in 1955. Transitive Algebra Problem has a close relationship with (hyper)invariant subspace. An affirmative answer to the transitive algebra problem would imply an affirmative answer to (hyper)invariant subspace problem. Meanwhile, the research to the transitive algebra prob-lem impact the operator theory very much. Transitive algebra problem, in which field many mathematicians have mad a lot of work since the problem was raised, is an important issue in the operator theoryIn the first chapter of this article, the transitive algebra problem, (hyper)invariant sub-space problem and the relationship between them are presented. And then we introduce the Burnside theorem that states B(H) is the only transitive algebra on the finitely dimensional Hilbert space.In the second chapter, we list some partial solutions to the transitive algebra problem which include the work of Arveson, Lomonosov, Nordgren, Radjavi and others. Most of them has the form: the problem could be answered if the transitive algebra satisfies some additional conditions.In the last chapter, we give some examples about the transitive algebra and (hyper)invari-ant subspace by the theory and the tool we introduced, and about the generator of B(H)...