Most of the structure theorems for complex matrices can be expressed solely in terms of invariant subspaces. For example, the statement that every nxn complex matrix is unitarily equivalent to an upper triangular matrix (from which the spectral theorem for normal matrices easily follows) is equivalent to the existence of a chain of invariant subspaces having one of each possible dimension from 0 to n. A matrix is similar to a single Jordan block if and only if its lattice of invariant subspaces is a chain; this allows for the Jordan form to be expressed in terms of invariant subspaces. If you look to infinite-dimensional Hilbert spaces, the sub-Hilbert spaces are closed linear subspaces, and the natural analogue of matrix is a bounded linear operator. If you want to extend the finite-dimensional structure theory to the infinite-dimesnional situation the first natural question to ask is whether every operator has a nontrivial (closed, linear) invariant subspace. This problem was popularized by Paul Halmos in the 1970's and, while the solution may not be important, attempts at solutions have generated a vast amount of important mathematics. For example, the concept of quasidiagonality for C*-algebras, which is very important to that subject, was defined by Halmos as a reducing version of quasitriagularity (a property distilled from several theorems about the invariant subspace problem).