# Download Analytic Hilbert modules by Xiaoman Chen, Kunyu Guo PDF By Xiaoman Chen, Kunyu Guo

The seminal 1989 paintings of Douglas and Paulsen at the idea of Hilbert modules over functionality algebras brought on a few significant examine efforts. This in flip resulted in a few fascinating and necessary effects, fairly within the parts of operator conception and practical research. With the sector now commencing to blossom, the time has come to gather these ends up in one quantity. Written through of the main energetic and often-cited researchers within the box, Analytic Hilbert Modules deals a transparent, logical survey of contemporary advancements, together with advances made through authors and others. It offers much-needed perception into functionality idea of a number of variables and comprises major effects released right here for the 1st time in components resembling attribute house idea, pressure phenomena, the equivalence challenge, Arveson modules, extension concept, and reproducing Hilbert areas on n-dimensional complicated house.

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Extra info for Analytic Hilbert modules

Example text

Then each module map σ : M1 → M2 is canonical. In fact, from [DY1], one knows that there exists a function φ ∈ L∞ (Tn ) (or φ ∈ L∞ (∂Ω)) such that for each h ∈ M1 , σ(h) = φh. One can extend φ to a meromorphic function φ˜ on Dn (or Ω) and φ˜ is analytic on Dn \ Z(M1 ) (or Ω \ Z(M1 )) such that for each z ∈ Dn \ Z(M1 ) (or z ∈ Ω \ Z(M1 )), ˜ h(z). From the above discussion one sees that the map σ is σ(h)(z) = φ(z) canonical. 4 Recall that a submodule M of X is said to have the codimension 1 property if dim M/[Uz M ] = 1 for each z ∈ Ω \ Z(M ), where Uz = {p ∈ C : p(z) = 0}.

His explicit characterization for all submodules of the Hardy module H 2 (D), in terms of the inner–outer factorization of analytic functions, has had a major impact. Since the Hardy module over the unit disk is the primary nontrivial example for so many different areas, it is not surprising that this characterization has proved so important. Almost everyone who has thought about this topic must have considered the corresponding problem for higher dimensional Hardy module H 2 (Dn ). Although the existence of inner functions in the context is obvious, one quickly sees that a Beurling-like characterization is not possible [Ru1], and hence this directs one’s attention to investigating equivalence classes of submodules of analytic Hilbert modules in reasonable sense.

In a unique factorization domain, every minimal prime ideal is principal [ZS, Vol(I) p. 238]. Since GCD{p1 , p2 , · · · , pk } = 1, the associated prime ideals J1 , J2 , · · · , Jn must all be maximal, and hence each Js must have the form (z − zs , w − ws ) with (zs , ws ) ∈ C2 , s = 1, 2, · · · , n mutually different. Therefore we can choose an integer, say m, sufficiently large such that Jsm = (z − zs , w − ws )m ⊂ Is for each s. Then ∩ns=1 Jsm ⊂ ∩ns=1 Is = (p1 , p2 , · · · , pk ). Obviously, the ideal ∩ns=1 Jsm is of finite codimension, and hence (p1 , p2 , · · · , pk ) is of finite codimension.