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.