A Brief Introduction to the Invariant Subspace Problem

The Invariant Subspace Problem asks if every bounded, linear operator on a complex Hilbert space has a non-trivial invariant subspace. In spite of its simple statement, this is one of the most famous unsolved problems on Operator Theory. We will present and skecth some of the most important techniques that have been developed to try to answer this question. In particular, we will discuss methods to provide non-trivial invariant subspaces for some classes of operators (Compact and Normal Operators) and methods to reduce the problem to the study of lattices of non-trivial invariant subspaces for concrete operators defined on spaces of analytic functions.