Lecture 1. Origins in Single Operator Theory
Many of the concepts that are now a basic part of this subject (triangulär
Operator algebras, compact perturbations, quasitriangular Operator algebras) have
identifiable origins in Single Operator theory. Others do not (e.g., the contents of
Lectures 4-8). In this lecture I want to describe some ideas from single Operator
theory that have led to significant generalizations in the theory of Operator
algebras. I will also discuss some ways of thinking about these things that I have
found to be useful.
Consider first the case of an Operator Ô on a finite-dimensional Hubert Space
3%*. Every graduate Student knows that there is an orthonormal basis
{el,e1,...,en} for ^relative to which the matrix of Ô is upper triangulär.
Equivalently, there exist (selfadjoint) projections 0 = P0 Ñë · · · Pn = 1
such that
0) (1-ÑÁ )ÃÑ
= 0 , 0 * é é,
(ii) thealgebrageneratedby {Pk} is maximalabelian.
The key step in the proof consists of showing that every Operator on a nonzero
finite-dimensional Hubert space has an eigenvector—a one-dimensional invariant
subspace. The triangulär form (1.1) is obtained by repeatedly applying that result
to the projections of Ô onto the various quotients
3e/pk^r, Ë = ï, é Ë - é,
as the
are constructed one by one. Nothing like this is known for Operators
on infinite-dimensional Hubert Spaces; indeed it is not even known if general
Operators have even a Single nontrivial invariant subspace.
In order to discuss this further, let us call an Operator Ô (or a set of Operators
{Ta\ á e /}) transitive if the only closed subspaces Jt of the underlying Hubert
space 3ýÑ satisfying
M c j ? (resp. TaJt c Jtfor all a)
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