Quotients of Standard Hilbert Modules

Details Quotients of Standard Hilbert Modules Quotients of Standard Hilbert Modules

2005-02-17

arxiv.org/abs/math/0502388v2

Mathematics

Operator Algebras

An extensive revision, rewritten from scratch with different main results. 33 pages

Scientific paper

We initiate a study of Hilbert modules over the polynomial algebra A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quotients by submodules give rise to universal solutions of nonlinear relations. Essentially all of the basic Hilbert modules that have received attention over the years are standard - including the Hilbert module of the d-shift, the Hardy and Bergman modules of the unit ball, modules associated with more general domains in complex d-space, and those associated with projective algebraic varieties. We address the general problem of determining when a quotient H/M of an essentially normal standard Hilbert module H is essentially normal. This problem has been resistant. Our main result is that it can be "linearized" in that the nonlinear relations defining the submodule M can be reduced, appropriately, to linear relations through an iteration procedure, and we give a concrete description of linearized quotients.

Affiliated with

Also associated with

No associations

Rating

Quotients of Standard Hilbert Modules does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quotients of Standard Hilbert Modules, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quotients of Standard Hilbert Modules will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-97515