Some geometric invariants from resolutions of Hilbert modules

Douglas, Ronald G.; Misra, Gadadhar; Varughese, Cherian

doi:10.1007/978-3-0348-8362-7_10

Cited by 5 publications

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“…Thus quasifree Hilbert modules can be characterized up to unitary equivalence by the curvature and a finite set of partial derivatives of curvature over Ω. In a series of papers [20], [22], [21], [23], the authors along with Verughese, have related the geometrical invariants for Hilbert modules in a resolution. In particular, one shows for the quotient module defined by the functions in a quasi-free module R that vanish to some order along a hypersurface, that the geometric invariants for the spectral sheaf for the quotient are determined by those for the quasi-free sheaf in the form of longitudinal curvature, transverse curvature and a second fundamental form involving an appropriate jet bundle.…”

Section: Usefulness Of Resolutionsmentioning

confidence: 99%

Quasi-Free Resolutions Of Hilbert Modules

Douglas

Misra

2003

Integral Equations and Operator Theory

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Abstract. The notion of a quasi-free Hilbert module over a function algebra A consisting of holomorphic functions on a bounded domain Ω in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra A. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module M is said to be compactly supported if there exists a constant β satisfying ϕf ≤ β ϕ X f for some compact subset X of Ω and ϕ in A, f in M. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.

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Section: Usefulness Of Resolutionsmentioning

confidence: 99%

Quasi-Free Resolutions Of Hilbert Modules

Douglas

Misra

2003

Integral Equations and Operator Theory

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“…Remark 2.11 (Resolutions of modules over function algebras). We also point out that our use of the terms resolution and free resolution differs substantially from usage of similar terms in work of Douglas, Misra and Varughese [DMV00], [DMV01], [DM03a], [DM03b]. For example, in [DM03b], the authors consider Hilbert modules over certain algebras A(Ω) of analytic functions on bounded domains Ω ⊆ C d .…”

Section: Statement Of Resultsmentioning

confidence: 99%

The free cover of a row contraction

Arveson

2004

Doc. Math.

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We establish the existence and uniqueness of finite free resolutions -and their attendant Betti numbers -for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory.For example, every graded d-contraction that is finitely multi-cyclic has a unique free cover of finite type -whose kernel is a Hilbert module inheriting the same properties. This contrasts sharply with what can be achieved by way of dilation theory (see Remark 2.5).

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Singular Hilbert Modules on Jordan–Kepler Varieties

Misra

Upmeier

2020

Operator Theory: Advances and Applications

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Some geometric invariants from resolutions of Hilbert modules

Cited by 5 publications

References 15 publications

Quasi-Free Resolutions Of Hilbert Modules

Quasi-Free Resolutions Of Hilbert Modules

The free cover of a row contraction

Singular Hilbert Modules on Jordan–Kepler Varieties

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