Download e-book for kindle: Analytic Hilbert modules by Xiaoman Chen, Kunyu Guo


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 triggered a few significant study efforts. This in flip ended in a few fascinating and useful effects, rather within the components of operator concept and practical research. With the sector now starting to blossom, the time has come to assemble these leads to one quantity. Written by means 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 by way of authors and others. It presents much-needed perception into functionality idea of numerous variables and contains major effects released right here for the 1st time in parts comparable to attribute area conception, pressure phenomena, the equivalence challenge, Arveson modules, extension conception, and reproducing Hilbert areas on n-dimensional complicated area.

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By Beurling’s theorem, each submodule of H 2 (D) is principal. Even more generally, for any analytic Hilbert module X on a domain in the complex plane C, a submodule generated by some polynomials is principal because every ideal of polynomials in one variable is principal. However for the Hardy module H 2 (Dn ) in several variables, there exist submodules with any rank [Ru1, p. 72]. Let λ ∈ Cn . We denote by Oλ the ring of all germs of analytic functions at λ. For detailed information about Oλ we refer the reader to [DY1, Kr].

Proof. We assume rank(M ) = m with a generating set {f1 , f2 , · · · , fm }. Now let P : M → M [IM ] be the orthogonal projection. We write f˜i = P fi for i = 1, 2, · · · , m. Since I annihilates the quotient module M/[IM ], one can consider M/[IM ] as a C/I-module with a generating set {f˜1 , f˜2 , · · · , f˜m }. Since C/I is finite dimensional, this ensures dim M/[IM ] < ∞, and actually dim M/[IM ] ≤ codim(I) rank(M ). 8 Let M be a finitely generated submodule of an analytic Hilbert module X, and let I be a finite codimensional ideal of C.

The following theorem is from [Guo1]. 1 Let X be an analytic Hilbert module and M a submodule of X. Then we have (1) if Z(M ) = ∅, then M e = X; (2) if Z(M ) = ∅, then M ⊆ M e = X, (M e )e = M e , and Z(M ) = Z(M e ); (3) if Z(M ) = ∅, then M e = λ∈Z(M ) Mλe . In particular, if M1 , M2 are two submodules of X, then M1e = M2e if and only if Z(M1 ) = Z(M2 ), and for every λ ∈ Z(M1 ), M1λ = M2λ . Note that (3) says that the AF-envelope of M is equal to the intersection of envelopes of M at all zero points.

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Analytic Hilbert modules by Xiaoman Chen, Kunyu Guo

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