Topics in Operator Theory (Mathematical Survey)

The five articles in this volume are expository in nature, and they all deal with various aspects of the theory of bounded linear operators on Hilbert space. The volume is very timely, because in the last year or two great progress has been made on hard problems in this field, and thus operator theory today is a very exciting part of mathematical research. One particular problem on which considerable progress has been made recently is the invariant subspace problem. This is the question whether every bounded linear operator on a separable, infinite-dimensional, complex Hilbert space $\mathcal H$ has a nontrivial invariant subspace.Even though this problem remains unresolved, there are some operators T on $\mathcal H$ for which the structure of a lattice of all invariant subspaces of T is even, and the first article in this volume, 'invariant subspaces' by Donald Sarason, is added to a discussion of such operators. One of the interesting features of this lucid presentation is the interplay between operator theory and classical analysis. The second article is entitled 'Weighted shift operators and analytic function theory' and was written by Allen Shields. He has taken essentially all of the information presently given about weighted shift operators (with scalar weights) and incorporated it into this comprehensive article.A central theme of the composition is the interaction between weighted shift operators and analytic function theory, and in an added bonus for the reader, the article contains a list of thirty-two interesting research problems. The third article in the volume is a treatise called 'A version of multiplicity theory' by Arlen Brown. The problem treated is how to decide when two normal operators are unitarily equivalent. (Unitary equivalence is the analog for operators of the concept of isomorphism for groups, rings, etc.) The unitary equivalence problem for arbitrary operators is exceedingly difficult, but the theory of spectral multiplicity, which can be approached in several different ways, furnishes a reasonable complete set of unitary invariants for normal operators.The author focuses attention on the concept of a spectral measure, and his clear presentation of this circle of ideas should lead to a better understanding of multiplicity theory by beginners and experts alike. The fourth article in this volume, 'Canonical models' by R. G. Douglas, is concerned with the theory of canonical models for operators on Hilbert space. The central underlying idea is that if T is any contraction operator on $\mathcal H$ (i.e., if the norm of T is at most 1), then there is a canonical construction that associates with T an operator $\mathrm{M}_\mathrm{T}$ that is unitarily equivalent to T, called its 'canonical model'.One can therefore study T by studying $\mathrm{M}_\mathrm{T}$ instead, and this theory has made significant progress in the past ten years. The author, who has contributed substantially to the geometrization of this theory, exposes in his article various important components of the theory, and thereby gives the reader much insight into its successes and failures. The final article in this volume, 'A survey of the Lomonosov technique in the theory of invariant subspaces' by Carl Pearcy and Allen Shields, is a survey of some new invariant-subspace theorems that resulted from the brilliant and elegant method of proof introduced by Victor Lomonosov early in 1973. Further study and refinement of this technique should lead to additional progress on the invariant subspace problem.