The notion of an invariant subspace is fundamental to the subject of operator theory. Given an operator T on a Banach space X, a closed subspace M of X is said to be a non-trivial invariant subspace for T if T(M) " M and M #= {0},X. A famous unsolved problem, called the “invariant subspace problem,” asks whether every bounded linear operator on a Hilbert space (more generally, a Banach space) admits a non-trivial invariant subspace. In this thesis, we discuss the greatest achievements in solving this problem for special classes of linear operators. We include several positive results for linear operators related to compact operators and normal operators, and negative results for certain linear operators on Banach spaces. Our goal is to build up the theory from the basics, and to prove the main results in a way that is accessible to a student who is relatively new to the world of functional analysis.
