By Xiaoman Chen, Kunyu Guo

The seminal 1989 paintings of Douglas and Paulsen at the thought of Hilbert modules over functionality algebras caused a few significant study efforts. This in flip ended in a few interesting and beneficial effects, relatively within the parts of operator conception and useful research. With the sphere now starting to blossom, the time has come to assemble these leads to one quantity. Written by means of 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 by means of authors and others. It offers much-needed perception into functionality conception of numerous variables and contains major effects released the following for the 1st time in parts equivalent to attribute area concept, tension phenomena, the equivalence challenge, Arveson modules, extension conception, and reproducing Hilbert areas on n-dimensional complicated house.

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**Example text**

Let σ : M1 → M2 be a module map and M1 have the codimension 1 property. Then it is easy to verify that the map σ is canonical. For the analytic Hilbert module X, we let A(Ω) denote the closure of all polynomials in the operator norm. Then A(Ω) is a Banach algebra consisting of analytic functions in X. It is easy to know that each C-module map extends uniquely to an A(Ω)-module map. 5 Let M1 , M2 be submodules of X, and M1 ∩ A(Ω) = {0}. If σ : M1 → M2 is a module map, then the map σ is canonical.

Z . zα 2 Define the conjugate linear map γ : C → C by γ( aα z α ) = a ¯α α! α z . zα 2 It is easy to verify that γ is one to one, and onto. Thus the image of M0 under the conjugate linear operator γ is a subspace of M ⊥ . 6, codim M = dim M0 . Therefore, we get that M ⊥ = γ(M0 ). Hence M ⊥ is a finite dimensional invariant polynomial space, completing the proof. Let M be a finite codimensional submodule of L2a (Ω). Then M has finitely many zero points λ1 , λ2 , · · · , λl in Ω such that M can be uniquely represented as l M= Mi , i=1 where Mi is a finite codimensional submodule and has a unique zero λi .

If I is finite codimensional, then by [AM], there exists a positive integer m such that √ √ ( I)m ⊆ I ⊆ I. 9 Let I1 , I2 be finite codimensional ideals with I1 = I2 . Then deg pI1 ,M = deg pI2 ,M . Proof. Obviously, we only need to prove that for any finite codimensional ideal I, deg pI,M = deg p√I,M . Since there exists a positive integer m such that √ √ ( I)m ⊆ I ⊆ I, we have p√I,M (k) ≤ pI,M (k) ≤ p(√I)m ,M (k) = p√I,M (mk) for large integer k. From the above inequalities, the desired result follows.