Every linear map on Cn has an upper triangular form; and for a fixed basis, the set of upper triangular matrices is a tractable object. For operators on Hilbert space, the notion of triangular form is replaced by the search for a maximal chain of invariant subspaces. This has been a rather intensive search, but the Invariant Subspace Problem remains, and is likely to remain for some time. The study of nest algebras takes the other point of view: fix a complete chain of closed subspaces (a nest) and study the algebra of all operators leaving each element of the nest invariant. That is, we study all operators with a given triangular form.

This sub-discipline of operator theory is about twenty five years old. It has reached a stage where there are many nice results, and a fairly satisfactory theory. Yet there are still interesting and compelling problems remaining. In these lectures, I will attempt to describe some of the results and to state some of these open questions.

Closely related to nest algebras are the so called CSL algebras. A CSL is a complete lattice L of commuting projections. The associated algebra Alg L consists of all operators leaving the ranges of L invariant. That is, all operators A such that P⊥-AP = 0 for all P in L.