Preface This volume consists of a collection of papers presented at the 1993 Summer Research Conference on Multivariable Operator Theory, held at the University of Washington in Seattle, under the auspices of the American Mathematical Society. The articles represent contributions to a variety of areas and topics, which may be viewed as forming an emerging new subject, involving the study of geometric rather than topological invariants associated with the general theme of operator theory in several variables. Beginning with J. L. Taylor's discovery in 1970 of the right notion of joint spectrum and analytic functional calculus for commuting families of Banach space operators, a number of significant developments have taken place. For instance, a Bochner-Martinelli formula has been generalized to commuting n- tuples in arbitrary C* -algebras various forms of the multivariable index theorem have been proved using non-commutative differential geometry and algebraic geometry and systems of Toeplitz and Hankel operators on Reinhardt domains, bounded symmetric domains, and domains of finite type have been substantially understood from the spectral and algebraic viewpoints, including the discovery of concrete Toeplitz operators with irrational index. These developments have been applied successfully to various types of quanti- zations, and functional spaces on Cartan domains and on pseudoconvex domains with smooth boundary have been thoroughly studied. A generalization of the Berger-Shaw formula to several variables has been proved and connections with the local multiplicative Lefshetz numbers, analytic torsion, and curvature rela- tions of canonically associated hermitian vector bundles have been established. Moreover, a sophisticated machinery of functional homological algebra suitable for the study of multivariable phenomena has been developed, and a rich theory for invariant pseudodifferential operators on domains with transverse symmetry has been produced. Much of multivariable operator theory involves the interaction between the subspace geometry of defect spaces and algebraic K-theory. For one example, a multivariable index theorem corresponding to commuting pairs of elements with finite defect A, B E End( H), where H is a vector space over an arbitrary field F, has emerged in connection with the Quillen algebraic K-theory. The ix
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