Factorization Of Generalized Holomorphic Curves And Similarity Of Operators

In 1978,M.J.Cowen and R.G.Douglas[10]defined a class of geometric operators:Cowen-Douglas operators.The unitary classification and similarity classification of op-erators in Bn(?)is one of the important topics in the study of operator classification.In 1978,M.J.Cowen and R.G.Douglas proved that the curvature and its covariant deriva-tive of holomorphic complex bundles are the unitary invariants of operators in Bn(?).In 2000,Kehe Zhu[44]characterized the unitary classification and similarity classification of operators of operators in Bn(?)by using spanning holomorphic cross-section.In 1984,G.Misra[35]described the homogeneity of operators T ? B1(D),using techniques of cur-vature of holomorphic complex bundles.In 2011,A.Koranyi and G.Misra[29]gave a complete characterization of the homogeneous operators T?Bn(D)by using complex geometry and group representation theory.In this dissertation,we use the tensor struc-ture of holomorphic vector bundles to give a new characterization of the homogeneous operators T?Bn(D).In 1997,M.Martin and N.Salinas[33]defined the holomorphic curves on C*-algebras(generalized holomorphic curves),using the relationship between holomorphic vector bun-dles induced by T?Bn(D)and C*-algebra.At the same time,the unitary classification of generalized holomorphic curves was given.It can be regarded as a generalization of Cowen-Douglas theory on C*-algebras.The generalized holomorphic curve theory has an important application in the study of similarity classification of Cowen-Douglas oper-ators.In 2009,H.Kwon and S.Treil[29]showed that a contraction operator T ? Bn(D)is similar to the(?)i-1nMz*if and only if ?(?)?T(?)?(?)22-n/(1-|w|2)2???(?),where Mz*is adjoint of the multiplication operator in the Hardy space H2,? is a bounded subharmonic function and ?T(?)is the orthogonal projection onto ker(T-?).Then it is not hard to see that ?T(?)is a typical generalized holomorphic curve.The beautiful theorem mainly involves the similarity between the contraction operator T?Bn(D)and(?)i=1n Mz*from the perspective of partial derivatives of generalized holomorphic curves.In this dissertation,we discuss the similarity classification of Cowen-Douglas operator with the spectrum ?in terms of the factorization of generalized holomorphic curve.The dissertation is divided into three parts.In the first part,we give a new charac-terization of the homogeneity of operators by using the tensor structure of holomorphic vector bundles.In the second part,the similarity classification of operators in Bn(?)is described by factorization of generalized holomorphic curves.In the third part,we calculate the curvature of some Cowen-Douglas operators for the specific spectral figures and characterize its similarity.