Hilbert Die Curvature And Local Dimension Formula

This thesis mainly concerns the relationship between Arveson's curvature in-variant and the index of the Dirac operator under the assumption that the Hilbert module is quasihomogeneous, the Taylor spectrum and Taylor essential spectrum of the so called Beurling type of submodules and their quotient modules of the Hardy module over the unit ball in higher dimension, together with their associated K-homology, and the dimension formula for the localization of analytic submod-ules generated by polynomials together with some calculations of their geometric invariants.In 1999, Arveson introduced the notion of curvature invariant of Hilbert mod-ules which is a complicated analytic invariant by the definition. Arveson has noticed that his curvature invariant is always an integer under some interesting cases. This remarkable fact maybe imply many properties of the Hilbert modules which are un-known, so it arose many mathematicians' attention and has been one of the problem of focus in multi-variate operator theory and geometric analysis of Hilbert modules. In this thesis we do some research on the properties of quasi-homogeneous Hilbert modules, moreover, we generalize the main result in [Ar6] which relates Arveson's curvature invariant to the index of the Dirac operator in the case of graded Hilbert modules to the case where the Hilbert module is quasi-homogeneous.It is well known that every invariant subspace for the coordinate multiplication operator of the Hardy space over the unit disk is generated by single inner function. In the language of Hilbert modules, the submodule if determined by single inner function, so the theory of inner function is very important to study the structure of the submodules. We study the Beurling type submodules of the Hardy module over the unit ball in higher dimension, that is, those submodules which are generated by inner functions. We completely describe Taylor spectrum and Taylor essential spec-trum of the Beurling type of submodules and their corresponding quotient modules of the Hardy module over the unit ball of higher dimension. Moreover, we study their associated K-homology.Cowen and Douglas introduced the tools of complex geometry into the study of operator theory in late seventies of last century. In recent years, Douglas and Misra and other mathematicians want to study the geometric invariants of Hilbert module by using methods of complex geometry, for example, to study the curva-ture invariant of the Hermitian holomorphic vector bundles related to some classes of Hilbert modules. The technique of localization is one of the powerful tools to study Hilbert modules. During the study of geometry of the localization of Hilbert modules, Douglas, Misra and Varughese opposed a conjecture which concerns the dimension formula for the localization of analytic Hilbert submodules generated by polynomials. By using the characteristic space theory which was introduced and developed by Guo, we turn the conjecture into an equivalent elementary problem in classical algebraic geometry. But it is nontrivial to solve such an problem. We prove the equivalent problem is true under general nature conditions by combining the methods in operator theory and algebraic geometry. Moreover, by finding some classic examples in algebraic geometry we show that there are exceptions to their conjecture and so our assumptions are necessary to make the conjecture true.