Multiplication Operators On Complex Bundles

In their classical paper[3],M.J.Cowen and R.G.Douglas intro-duced a special class Bn(?)of operators on the Hilbert space.For an operator T ? Bn(?),they defined a related Hermitian holomor-phic vector bundle ET There are many interesting relations between the properties of the Cowen-Douglas operators and the corresponding complex bundles.M.J.Cowen and R.G.Douglas had shown that the"curvature" K of the complex Bundle Ef is a key invariant to deter-mines whether the complex bundles Ef and Ef are congruent or not.The "curvature" K is too hard to deal with it.Scholars are looking for other ways to describe those relations between the Cowen-Douglas operators and the complex bundles.Professor CaoYang gives a more elementary unitary invariant of complex bundle in the paper[C].It is a equivalent class of the n X n complex functions matrix.Professor CaoYang and professor JiYouqing use it to consider the HIR decom-positions of the complex bundles and Cowen-Douglas operators and another related problems.From the above viewpoint,we will enlarge the operator class which we can deal with.It' s well known that a weighted shift operator T can be seen as a ordinary shift operator Mz on a Hilbert space of formal power series,that is,multiplication by z(cf.[12],p57-58).On the other hand,a Cowen-Douglas operator acts on complex bundle as multiplication operator on the fibres.For this reason,it is worth to study general multiplication operators acting on complex bundle.In this paper will study the basic properties on those facts.In chapter 1-7,we review some basic results from reference[C][CJ].After briefly reviewing a new unitary invariant for Cowen-Douglas operators by the complex bundle,and then we consider the HIR de?composition of Cowen-Douglas operators by using the new unitary invariant,and then we study the HIR decomposition of "pull back"complex bundle.In chapter 8,we focus on the following issues.Lem-ma 8.1.1 and proposition 8.1.1 give a necessary condition of local "pull back" bundle,then we give the concept of the discriminant matrix,and then we want to know that when a complex bundle is a“pull back" bundle of a holomorphic curve.In the therom 8.2.1,we draw on the lessons of the celebrated paper[1]of E.Calabi,we give that a complex bundle is a sufficient condition for a "pull back" bundle of a holomorphic curve.