On many approaches of the invariantsubspace problem in Hilbert spaces

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The invariant subspace problem is the question that every bounded operator on a separable Hilbert space has a non-trivial invariant subspace. There has been a large amount of work on invariant subspaces, motivated by interest in the structure of non-self adjoint operators on Hilbert space. Our purpose is to investigate this problem on many ways; theory of operator algebras, Banach spaces and Hilbert spaces. And we succeeded in development of much theory relevant to the problem.