Deconstructing Dirac Operators I: Quantitative Hartogs-Rosenthal Theorems

Martin, Mircea

doi:10.1142/9789812835635_0102

Cited by 7 publications

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“…By refining our tools it is possible to establish more general estimates for first-order differential operators with constant coefficients from a real or complex Banach algebra. This specific issue is addressed to a certain extent in Martin [13]. Other recent developments of this research project will be reported elsewhere.…”

Section: Aacamentioning

confidence: 98%

Uniform Approximation by Closed Forms in Several Complex Variables

Martin

2009

AACA

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The first main result in this article provides uniform estimates for solid Bochner-Martinelli-Koppelman transforms of type (p, n − 1), 0 ≤ p ≤ n, of continuous forms on a compact set Ω in the complex space C n , in terms of the Euclidean volume of Ω. In the single variable case this result generalizes a classical inequality for the Cauchy kernel due to Ahlfors and Beurling. The second main result is a quantitative Hartogs-Rosenthal theorem which points out that the uniform distance in the space of continuous (p, n − 1)-forms, 0 ≤ p ≤ n, on a compact set Ω in C n from a smooth form to the subspace of ∂-closed forms on Ω is controlled by the Euclidean volume of Ω. This theorem as well as a third main result are generalizations to higher dimensions of an inequality in single variable complex analysis due to Alexander.

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Section: Aacamentioning

confidence: 98%

Uniform Approximation by Closed Forms in Several Complex Variables

Martin

2009

AACA

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“…Since ∇ and D uniquely determine each other, the two Laplace operators on a Dirac bundle E are related by an equation of the form 25) referred to as the Bochner-Weitzenböck identity, where the remainder R is an operator of order zero that only depends on the curvature operator of E associated with the linear connection ∇. We next turn our attention to complex manifolds and complex vector bundles.…”

Section: Dirac and Laplace Operators On Dirac Vector Bundlesmentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C * -algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C * -algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

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Function Spaces in Quaternionic and Clifford Analysis

Martin¹

2015

Operator Theory

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Deconstructing Dirac Operators I: Quantitative Hartogs-Rosenthal Theorems

Cited by 7 publications

References 0 publications

Uniform Approximation by Closed Forms in Several Complex Variables

Uniform Approximation by Closed Forms in Several Complex Variables

Seminormal systems of operators in Clifford environments

Function Spaces in Quaternionic and Clifford Analysis

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