Existence of Invariant and Hyperinvariant Subspaces

Abstract

We have seen that certain operators have non-trivial invariant subspaces: normal operators (Corollary 1.17), operators with disconnected spectra (Corollary 2.11), parts of the adjoint of the unilateral shift (Corollary 3.31), and operators closely related to compact operators (Corollaries 5.5, 5.6, and 5.7). In this chapter we show that operators that are close (in a certain sense) to normal operators with thin spectra have invariant subspaces. In particular, we obtain the result that parts of the adjoints of finite-multiplicity unilateral shifts have invariant subspaces, thereby proving the factorization theorem for isometry-valued analytic functions alluded to in Section 3.6.