Invariant Subspaces Of Normal - Like Operators

There is an outstanding problem in operator theory, the so-called, INVARIANT SUBSPACE PROBLEM: Given a complex Banach space X, which operators on X have non-trivial closed invariant subspaces? This problem has been open for more than half a century. In spite of momentous efforts by functional analysts, the problem continues to elude them even today. Until fairly recently, it was not known whether there was any operator T without a non-trivial closed invariant subspace.