Joint seminormality and Dirac operators

Abstract

In this article, an approach to joint seminormality based on the theory of Dirac and Laplace operators on Dirac vector bundles is presented. To eachn-tuple of bounded linear operators on a complex Hilbert space we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of thatn-tuple. Employing standard spin geometry techniques we next get a Bochner type and two Bochner-Kodaira type identities in multivariable operator theory. Further, four different classes of jointly seminormal tuples are introduced by imposing semidefiniteness conditions on the remainders in the corresponding Bochner-Kodaira identities. Thus we create a setting in which the classical Bochner's method can be put into action. In effect, we derive some “vanishing theorems” regarding various spectral sets associated with commuting tuples. In the last part of this article we investigate a rather general concept of seminormality for self-adjoint tuples with an even or odd number of entries.