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

Á

= 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

Pk9s

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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http://dx.doi.org/10.1090/cbms/055/01