On invariant subspaces of several variable Bergman spaces

Abstract

0. Introduction. Some recent investigations have been concerned with the structure and classification of the invariant subspaces of the Bergman n-tuple of operators, cf. Agrawal-Salinas [2], Axler-Bourdon [4], Bercovici [5], Douglas [7], Douglas-Paulsen [8]. Due to the richness of this lattice of invariant subspaces, the additional assumption on finite codimension was naturally adopted by the above mentioned authors as a first step towards a better understanding of its properties. The present note arose from the observation that, when the Lbounded evaluation points of a pseudoconvex domain lie in the Fredholm resolvent set of the associated Bergman rc-tuple, then the description of finite codimensional invariant subspaces is, at least conceptually, a fairly simple algebraic matter. This simplification requires only the basic properties of the sheaf model for systems of commuting operators introduced in [11]. The main result below is also available by some other recent methods. First is the quite similar technique of localizing Hubert modules over function algebras, due to Douglas [7] and Douglas and Paulsen [8], and secondly is the study of the so-called canonical subspaces of some Hubert spaces with reproducing kernels, developed by Agrawal and Salinas [2]. Both points of view will be discussed in §2 of this note. In fact the Bergman space of a pseudoconvex domain is only an example within a class of abstract Banach ^(C^-modules, whose finite codimensional submodules turn out to have a similar structure. The precise formulation of this remark ends the note. We would like to thank the referee, whose observations pointed out some bibliographical omissions in a first version of the manuscript.