On the Rigidity of Bergman Submodules

Abstract

Introduction. The search for and the classification of invariant subspaces of various operators acting on function spaces have proved to be two very rewarding research problems in operator theory and harmonic analysis. Not only that the questions themselves have turned out to be important, but even more so did they applications and the techniques used to solve them. For instance the classical theorem of Beurling [3] on the structure of analytically invariant subspaces of the Hardy space on the torus has been influential in many areas of modern mathematics, ranging from the dilation theory of a contraction, to interpolation problems in function theory or to the probabilistic analysis of time series. In Hilbert spaces of analytic functions depending on several complex vari? ables, such a simple classification as that of Beurling's theorem is no longer to be expected. Even the existence of inner functions can be a difficult question. The case of a Hardy space supported by a polydisk in Cn has been, however, successfully studied during the last two decades, see [12], [1]. More recently, R.G. Douglas and his collaborators have developed a general framework for investigating invariant subspaces from the point of view of Hilbert modules over function algebras, [5], [6], [7]. The importance ofthe algebraic language of mod? ules and their operations lies in its being deeply rooted into the structure theory of algebras of analytic functions. Thus the comparison between Hilbert modules and, say, sheaves of ideals of analytic functions becomes transparent and benefic. In this way Douglas and Paulsen [5] have isolated from a series a previous works a phenomenon which is specific only to multidimensional Hardy submodules: once two such submodules are analytically isomorphic they are equal, see [1], [5], [6] and [7] for the^precise statements. This rigidity behaviour of Hardy submodules contrasts with Beurling's theorem-the latter implying that all invari? ant subspaces of the Hardy space on the torus are unitarily equivalent. When compared with the purely algebraic case of ideals of a regular, Noetherian local ring, this anomaly is no longer surprising. The transition from a Hardy submodule to an ideal of a local ring, which is necessary in any proof of a rigidity result as before, is made by a localization functor, see [6].