JOINT SEMINORMALITY AND DIRAC OPERATORS

In this article, an approach to joint seminormality based on the theor y of Dirac and Laplace operators on Dirac vector bundles is presented. To each n-tuple of bounded linear operators on a complex Hilbert spac e we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of that n-tuple. Employing standar d spin geometry techniques we next get a Bochner type and two Bochner- Kodaira type identities in multivariable operator theory. Further, fou r different classes of jointly seminormal tuples are introduced by imp osing semidefiniteness conditions on the remainders in the correspondi ng Bochner-Kodaira identities. Thus we create a setting in which the c lassical Bochner's method can be put into action. In effect, we derive some ''vanishing theorems'' regarding various spectral sets associate d with commuting tuples. In the last part of this article we investiga te a rather general concept of seminormality for self-adjoint tuples w ith an even or odd number of entries.