Cowen-douglas Operator With The Proposed Geometry Of The Free Hilbert Mode Theory

In this thesis, we mainly consider geometric theory of Cowen-Douglas oper-ators and Hilbert modules. We consider the characterization of the commutant and minimal reducing subspaces of Cowen-Douglas operators, unitary equivalence of quasi-free Hilbert modules and geometric theory of bundle shifts.In Chapter One, we study the geometric theory of Cowen-Douglas operators. For a given Cowen-Douglas operator, let V*(T) be the von Neumann algebra con-sisting of operators commuting with both T and T*. We identify operators in V*(T) with connection-preserving bundle maps on E(T), the holomorphic Hermitian vector bundle associated to T. By studying such bundle maps, the structure of V*(T) and minimal reducing subspaces of T can be determined in terms of geometric invariants.In Chapter Two, we consider unitary equivalence of quasi-free Hilbert modules. We complement the work of Douglas and Misra by showing that if the modulus function of two quasi-free Hilbert modules is the absolute value of a holomorphic bundle map, then these two quasi-free Hilbert modules are unitarily equivalent. In addition, we give geometric characterization of unitary equivalence in terms of dual bundles.In Chapter Three, we consider geometric theory of the bundle shift for a flat unitary vector bundle. Given a flat unitary bundle E, the adjoint of bundle shift TE is a Cowen-Douglas operator, which thus determines a vector bundle E(TE). The problem of determining the relationship of E and E(TE)is raised by Douglas in [Dou]. We will study the operator theory of the bundle shift from a geometric viewpoint, and give explicit forms of reducing subspaces of the bundle shift. Finally, we show that there exists a dual pairing between E and E(TE), and the dual paring is compatible with the holomorphic Hermitian structures of the two bundles. Combing with our results obtained in Chapter One, we exhibit how the holonomy group of E(TE) is related to E.