Ronald G. Douglas

Hilbert Modules over Function Algebras

Much of the early motivation for the study of operator theory came from integral equations although early in this century both operator theory and functional analysis took on a life of their own. Self-adjoint operators, both bounded and unbounded, occupied center stage for several decades either singly or in algebras. During the last two or three decades various approaches to the non-selfadjoint theory have been introduced with considerable success at least in the case of a single operator. The generalization to several operators, whether commuting or non-commuting, has largely eluded us. In this note we want to outline a different point of view which may assist in guiding developments in this area.

Adiabatic limits of the η-invariants the odd-dimensional Atiyah-Patodi-Singer problem

We study the η-invariant of boundary value problems of Atiyah-Patodi-Singer type. We prove the formula for the spectral flow of the families overS1. Assuming a product structure in a collar neighbourhood of the boundary, we show that the η-invariant behaves the same way as on a closed manifold. We also study the “adiabatic” limit of the spectral invariants. In nice cases we are able to split them into a contribution from the interior, one from the cylinder, and an error term. Then we show that the error term disappears with the increasing length of the cylinder.

Operator theory in the Hardy space over the bidisk (II)

In this paper, we identify the vector valued Hardy space with the Hardy space over the bidisk and construct a universal model for thecontractive analytic functions. We will also study some elementary properties of the submodules and show, in some cases, how the operator theoretical properties are related to the module theoretical properties. The last part focus on the study of double commutativity of compression operators.

Operator theory and algebraic geometry

On presente des resultats en theorie des operateurs multivariable dont les demonstrations se relient a des techniques issues de la geometrie algebrique

A harmonic analysis approach to essential normality of principal submodules

Guo and the second author have shown that the closure [I] in the Drury–Arveson space of a homogeneous principal ideal I in C[z1,…,zn] is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p>n. The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space XI of the resulting C⁎-algebra for the quotient module is shown to be contained in Z(I)∩∂Bn, where Z(I) is the zero variety for I, and to contain all points in ∂Bn that are limit points of Z(I)∩Bn. Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.

Cyclic cocycles, renormalization and eta-invariants

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Toeplitz Operators and Poincaré Duality

The study of matrices constant on all diagonals was introduced by Toeplitz [30]. Such matrices can be finite, semi-finite, or doubly-infinite. Aside from the profundity of results which have been obtained about such matrices the other amazing thing about them is the extent to which they occur in widely varied parts of mathematics, both pure and applied. Although several heuristic or even philosophical reasons could be advanced for this, we offer just one in this note and we concentrate entirely on the infinite cases.

Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂm is a bounded domain. LetM0 ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M0 The invariants are given explicitly in the particular case ofk = 2.

Geometric Invariants for Resolutions of Hilbert Modules

The development of complex function theory beyond the one-variable case required new techniques and approaches, not just an extension of what had worked already. The same is proving true of multi—variate operator theory. Still it is reasonable to start by seeking to understand in the larger context results that have proved important and useful in the study of single operator theory. For that reason the first author showed [6] how to frame the canonical model theory of Sz.-Nagy and Foias for contraction operators in the language of Hilbert module resolutions over the disk algebra. This point of view made clear why a straightforward extension of model theory failed in the multi—variate case. Since the appropriate algebra in this case would be higher dimensional, one would expect module resolutions, if they existed, to be of longer length and hence not expressible as a canonical model. While work on this topic has shed light on what one might expect to be true (cf. [7]), useful results are still scarce.

Topics in modern operator theory

On Closed Operator Algebras Generated by Analytic Functional Calculi.- A Conjecture Concerning the Pure States of B(H) and a Related Theorem.- A C*-Algebra Approach to the Cowen-Douglas Theory.- On Periodic Distribution Groups.- On the Smoothness of Elements of Ext.- Triviality Theorems for Hilbert Modules.- Exact Controllability and Spectrum Assignment.- Generalized Derivations.- Commutants Modulo the Compact Operators of Certain CSL Algebras.- Similarity of Operator Blocks and Canonical Forms. II. Infinite Dimensional Case and Wiener-Hopf Factorization.- Unitary Orbits of Power Partial Isometries and Approximation by Block-Diagonal Nilpotents.- Isomorphisms of Automorphism Groups of Type II Factors.- A Spectral Residuum for Each Closed Operator.- Two Applications of Hankel Operators.- A Rohlin Type Theorem for Groups Acting on von Neumann Algebras.- Derivations of C*-Algebras which Are Invariant Under an Automorphism Group.- Remarks on Ideals of the Calkin-Algebra for Certain Singular Extensions.- Modelling by L2-Bounded Analytic Functions.- The Maximal Function of Doubly Commuting Contractions.- Remarks on Hilbert-Schmidt Perturbations of Almost - Normal Operators.- Derivation Ranges: Open Problems.