Some Geometric Properties For Cowen-Douglas Operators

Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. In this paper, we define operations on the set of cross-sections dominated by a spanning holomorphic cross-section of E_T, by which the set is shown to be an algebra. Moreover, we prove this algebra is isomorphic to the commutant of T. Also, we prove the following results. If T₁, T₂∈B_n(Ω),γ₁ is the spanning holomorphic cross-section of E_T1, then T₁～T₂ if and only if there exists an operator X∈G(H), such that X_γ1 is the spanning holomorphic cross-section of E_T2. Moreover, we give a similarity classification of Cowen-Douglas operators.This paper includes four chapters. In chapter 1, we introduce the relative background in this paper and give some skeleton expressions of the original work. In chapter 2, we introduce some results and concepts used in this paper. In chapter 3, by some results of chapter 2, we get the main result of this paper. Moreover, we give a similarity classification of Cowen-Douglas operators. In chapter 4, we conclude the prime conclusion of this paper.