Notes on Invariant Subspaces

The main purpose of this article is to give an approach to the recent invariant subspace theorem of Brown, Chevreau and Pearcy: Every contraction on a Hubert space, whose spectrum contains the unit circle has nontrivial invariant subspaces. Our proof incorporates several of the recent ideas tying together function theory and operator theory. 1. I N T R O D U C T I O N The Jordan structure theorem for finite matrices has been known now for over one hundred years, and its usefulness can hardly be overstated. It says that every square matrix A over the complex numbers C is similar to another matrix B (i.e., B = XAX~ for some invertible matrix X) which is a direct sum of Jordan cells. That is, B can be written in block form \BX 0 ••• 0 ] B=\° * '" ° [o 0 ... Bk\ and each Bt has the form [A, 0 0