By Xiaoman Chen, Kunyu Guo

The seminal 1989 paintings of Douglas and Paulsen at the idea of Hilbert modules over functionality algebras induced a few significant examine efforts. This in flip resulted in a few exciting and invaluable effects, quite within the parts of operator idea and sensible research. With the sphere now starting to blossom, the time has come to assemble these leads to one quantity. Written by way of of the main lively and often-cited researchers within the box, Analytic Hilbert Modules bargains a transparent, logical survey of contemporary advancements, together with advances made via authors and others. It offers much-needed perception into functionality thought of a number of variables and contains major effects released right here for the 1st time in parts similar to attribute house idea, tension phenomena, the equivalence challenge, Arveson modules, extension conception, and reproducing Hilbert areas on n-dimensional advanced house.

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**Example text**

2003 CRC Press LLC Recall that the most simple submodules of H 2 (D) are p H 2 (D), where p are polynomials with Z(p) ⊂ D. Since each ideal I of polynomials in one variable is principal, this implies that [I] = p H 2 (D) for some polynomial p with Z(p) ⊂ D. The above view suggests that we consider submodules of analytic Hilbert module generated by ideals of polynomials. Equivalently, can we characterize submodule M which has the property that the ideal M ∩ C is dense in M [DPSY, Pau1]? Note that if M is as above and we let I = M ∩ C, then [I] ∩ C = I.

The proof is complete. 5 Let I = pL be the Beurling form of the ideal I. If every algebraic component of Z(I) has a nonempty intersection with Ω, then [I] is principal if and only if I = pC. In particular, if I is prime, and Z(I) ∩ Ω = ∅, then [I] is principal if and only if I = pC and p is prime. Proof. 4, Z(L) ∩ Ω = ∅. So for each λ ∈ Ω, [I]λ = [pC]λ . 3 we see that I ⊃ pC, and hence I = pC. The opposite direction is obvious. When n = 2, one can obtain a more detailed result. 6 Let X be an analytic Hilbert module on Ω(⊂ C 2 ), and let I = pL be the Beurling form of I.

In this section, we will examine some special cases in which the canonical model is unique. Using module theoretic language, this means that quotient modules are necessarily equal if they are unitarily equivalent (similar or quasi-similar). Let M1 and M2 be two submodules of analytic Hilbert module X, and M1 ⊇ M2 . Then we form the quotient module M1 /M2 on the ring C. The module action is given by p · (h + M2 ) = ph + M2 . It is easy to see that this action is bounded. In the following we will give a coordinate free method to deal with rigidity of quotient modules.