Analytic Hilbert modules by Xiaoman Chen, Kunyu Guo

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 learn efforts. This in flip resulted in a few fascinating and beneficial effects, rather within the parts of operator thought and useful research. With the sector now commencing to blossom, the time has come to assemble these leads to one quantity. Written via 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 by means of authors and others. It offers much-needed perception into functionality thought of numerous variables and comprises major effects released right here for the 1st time in components resembling attribute area concept, pressure phenomena, the equivalence challenge, Arveson modules, extension thought, and reproducing Hilbert areas on n-dimensional advanced house.

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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.

Then I is contracted in H 2 (D2 ) if and only if every algebraic component of Z(I) has a nonempty intersection with D2 . ©2003 CRC Press LLC Proof. We shall merely indicate that if I is contracted in H 2 (D2 ), then every algebraic component of Z(I) has a nonempty intersection with D2 . Let I = p L be the Beurling form of I. We decompose p as the product of the powers of irreducible polynomials, say, p = pk11 · · · pkl l . 9, L is of finite codimension and, hence, L has finitely many zeros, say, λ1 , · · · , λs .

This implies that for any polynomial p, the Toeplitz operator Tp¯ maps C to C. Now let P be a linear space consisting of polynomials. We say that P is an invariant polynomial space, if for any polynomial p, P is invariant under the action of Tp¯. It is easy to see that in the case of n = 1, an invariant polynomial space with the dimension m (1 ≤ m ≤ ∞) is the linear space with the basis {1, z 1 , · · · , z m }. 1 Let M be a finite codimensional submodule with a unique zero λ = 0. Then M ⊥ is a finite dimensional invariant polynomial space.

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