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Get Analytic Hilbert modules PDF

By Xiaoman Chen, Kunyu Guo

ISBN-10: 0203008839

ISBN-13: 9780203008836

ISBN-10: 1584883995

ISBN-13: 9781584883999

The seminal 1989 paintings of Douglas and Paulsen at the thought of Hilbert modules over functionality algebras prompted a few significant learn efforts. This in flip resulted in a few exciting and worthwhile effects, quite within the components of operator concept and useful research. With the sector now commencing to blossom, the time has come to gather 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 modern advancements, together with advances made through authors and others. It presents much-needed perception into functionality thought of numerous variables and comprises major effects released right here for the 1st time in parts reminiscent of attribute area thought, pressure phenomena, the equivalence challenge, Arveson modules, extension concept, and reproducing Hilbert areas on n-dimensional advanced house.

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1) Write M1 = M2 ⊕ R and restrict {Mz1 , Mz2 , · · · , Mzn } on R. By [Cur1], they can be simultaneously triangularized as   (1) λi   .. , Mzi =  .   (k) λi where i = 1, 2, · · · , n, and k = dim M1 /M2 , so that σp (Mz1 , Mz2 , · · · , Mzn ) is equal to {λ(1) , · · · , λ(k) }. 5, we have that σp (Mz1 , Mz2 , · · · , Mzn ) ⊂ Ω. ©2003 CRC Press LLC Writing Uλ(i) = {p : p are polynomials, and p(λ(i) ) = 0} i = 1, · · · , k, then Uλ(k) · · · Uλ(2) Uλ(1) M1 ⊆ M2 ⊆ M1 . Therefore, for λ ∈ Ω, but λ ∈ / σp (Mz1 , Mz2 , · · · , Mzn ), one sees that M1λ = M2λ .

2003 CRC Press LLC Chapter 3 Rigidity for analytic Hilbert modules The classic paper of Beurling [Beu] led to a spate of research in operator theory, H p -theory and other areas which continues to the present. 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.

2 Let I1 and I2 be ideals of polynomials such that each algebraic component of their zero varieties meets Ω nontrivially. If X/[I1 ] and X/[I2 ] are quasi-similar, then I1 = I2 . For the analytic Hilbert module X, we define the multiplier ring M (Ω) of X by M (Ω) = {f ∈ Hol(Ω) : f x ∈ X, ∀ x ∈ X}. It is easily checked that X ⊇ M (Ω) ⊇ A(Ω) ⊇ C. 3 Let M1 and M2 be two M (Ω)-submodules of X generated by multipliers. If X/M1 and X/M2 are quasi-similar over M (Ω), then M1 = M2 . ©2003 CRC Press LLC Proof.

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Analytic Hilbert modules by Xiaoman Chen, Kunyu Guo


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