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