Seminormal systems of operators in Clifford environments

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-Weitzenbock 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.