Contractive Hilbert Modules and Their Dilations over Natural Function Algebras

In this note, we show that quasi-free Hilbert modules R defined over natural function algebras satisfying a certain positivity condition, defined via the hereditary functional calculus, admit a dilation (actually a co-extension) to the Hardy module over the polydisk. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. Some consequences of this basic fact is then explored in the case of several natural function algebras. Introduction One of the most far-reaching results in operator theory is the fact that all contraction operators have a unitary dilation and a closely related co-isometric extension on which the model theory of Sz.-Nagy and Foias [11] is based. This model provides not only a theoretical understanding of the structure of contractions but provides a useful and effective method for calculation. A key reason this model theory is so incisive is the relatively simple structure of isometries due to von Neumann [12] . In particular, every isometry is the direct sum of a unitary and a unilateral shift operator defined on a vector-valued Hardy space. And, if one makes a modest assumption about the behavior of the powers of the contraction, then the unitary is absent and the co-isometry involved in the model is defined on a vector-valued Hardy space on the unit disk. If one attempts to extend this theory to commuting n-tuples of contractions on Hilbert space, then one quickly runs into trouble, particularly for the case of m > 2 in which case the example of Parrott [13] shows that a unitary or co-isometric extension need not exist. For the m = 2 case, Ando’s Theorem [1] seems to hold hope for a model theory since a pair of commuting contractions is known to have a unitary dilation. However, such dilations are not necessarily unique. But, more critically, the structure of the pair of commuting co-isometries is not simple and, in particular, need not be related to the Hardy space on the bidisk. In this note, we study the question of which commuting pairs (and m-tuples) of contractions have a co-isometric extension to backward shifts for the Hardy space for the bidisk (or polydisk). We don’t take up the question in this level of generality but assume that the commuting contractions are defined on a reproducing kernel Hilbert space. We approach the issue in the context of Hilbert modules over the algebra of polynomials using the notion of module tensor product. Since the latter is defined as a quotient of Hilbert modules, one can view the quotient realization of the module tensor product as a short exact resolution 2000 Mathematics Subject Classification. 47A13, 47A20, 46E20, 46E22, 46M20, 47B32.