Spin Geometry, Clifford Analysis, and Joint Seminormality

Martin, Mircea

doi:10.1007/978-3-0348-7838-8_12

Cited by 7 publications

(4 citation statements)

References 33 publications

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“…Significant early contributions to the development of the theory of joint seminormality, which prompted us to try and find a unifying framework, are due to Athavale [1], Cho, Curto, Huruya, and Zelazko [7], Curto [11], Curto and Jian [12], Curto, Muhly, and Xia [13], Douglas, Paulsen, and Yan [14], Martin and Salinas [29], McCullough and Paulsen [31], and Xia [44]. The viewpoint emphasized in our article was already outlined in Martin [19,[22][23][24] and is based on some techniques from Clifford analysis and spin geometry.…”

Section: Introductionmentioning

confidence: 93%

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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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Section: Introductionmentioning

confidence: 93%

“…For more details in this regard we refer to Martin [23,34]. However, the basic features of the models can be fully illustrated by assuming that each of the spaces H(x), x ∈ Ω, is one-dimensional, i.e., by taking the Lebesgue space H = L 2 (Ω), and that is exactly what we will be doing in this section.…”

Section: Riesz Transforms and Joint Hyponormalitymentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

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“…For more applications of algebra environments related to harmonic analysis and multivariable operator theory, we direct the attention of our reader to two groups of articles, Martin [36][37][38][39], and Martin [40][41][42][43]45], Martin, Salinas [50]. The list of specific issues includes Dirac operators with coefficients in a C *algebra, Cauchy-Pompeiu and Bochner-Martinelli-Koppelman representation formulas in a Banach algebra setting, maximal and fractional integral operators in Clifford analysis, generalizations of Ahlfors-Beurling and Alexander inequalities, quantitative Hartogs-Rosenthal theorems, Bochner-Weitzenböck and Bochner-Kodaira-Nakano self-commutator identities, extensions of Putnam inequality and singular integral models of seminormal systems of operators using Riesz transforms.…”

Section: Concluding Commentsmentioning

confidence: 99%

Algebra Environments I.

MARTIN

2024

Rev. Roumaine Math. Pures Appl.

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Algebra environments capture properties of non–commutative conditional expectations in a general algebraic setting. Their study relies on algebraic geometry, topology, and differential geometry techniques. The structure algebraic and Banach manifolds of algebra environments and their Zariski and smooth tangent vector bundles are particular objects of interest. A description of derivations on algebra environments compatible with geometric structures is an additional issue. Grassmann and flag manifolds of unital involutive algebras and spaces of projective compact group representations in C∗–algebras are analyzed as structure manifolds of associated algebra environments.

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“…To make the point, we would like to refer to Martin [10][11][12], where a real Clifford analysis approach addressing similar issues was developed. That approach uses several real variables and provides generalizations of both Alexander's and Ahlfors-Beurling's inequalities similar to Theorems A, B, and C to the setting of Clifford analytic functions, Dirac operators, and the Euclidean Cauchy kernel.…”

Section: Aacamentioning

confidence: 99%

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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Spin Geometry, Clifford Analysis, and Joint Seminormality

Cited by 7 publications

References 33 publications

Seminormal systems of operators in Clifford environments

Seminormal systems of operators in Clifford environments

Algebra Environments I.

Uniform Approximation by Closed Forms in Several Complex Variables

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