On The Unit Ball Of Hilbert Modules And K-cohomology

The thesis mainly concerns the essential normality of Hilbert modules over the unit ball, and its related geometry and analysis.In recent papers, Arveson considered Symmetric Fock space and its submodules, which are the theoretical basis of non-commutative dynamics system. The geometry and analysis on quotient submodules of Symmetric Fock space provide the concrete examples for algebraic geometry and algebraic topology, and excite the development and application of modern operator theory and operator algebra. A principle problem in Hilbert module is Arveson's conjecture, that is, the homogenous submodule is essentially normal. Douglas also arised the similar question in the Bergman module. Moreover, they conjectured that the K-homology element, which is a Hilbert module invariant given by the quotient module, is non-trivial.In this thesis, it is shown Arveson's conjecture is consistent with Douglas's conjecture. We also investigate the relation between the common zero set of the essentially normal submodule and the essential spectrum of its quotient submodule.We generalize Guo's work [Guo9] to the case of unitarily invariant Hilbert modules. It is shown the homogenous submodule is p-essentially normal in the case d = 2. In the case d > 3, some special cases are proved. Using different methods, we show that in the case d = 2, the quasi-homogenous submodule is p-essentially normal.In the case d = 2, we give an explicit expression of the module invariant, which show that it is non-trivial. In the case d = 3, the K-homology element is also non-trivial if the quotient is essentially normal.