Research On Similarity And Curvature Of Cowen-Douglas Operators

The unitary equivalence and similarity equivalence is a fundamental problem of the operator theory, looking for the completely similar invariants of the operator is one of the core problems of operator theory, but it is almost impossible to find the completely similar invariant of any bounded linear operator, so people only to consider the relatively special operator class. Although M. J. Cowen and R. G. Douglas have proved that the curvature function is unitary invariant of the Cowen-Douglas operators, but we don't know what is the relationship between curvatures and similarity equivalence of CowenDouglas operators, so this paper is about the study of the relationship between the curvatures and the similarity of Cowen-Douglas operators. This paper is to describe the similarity of Cowen-Douglas operators with index one by considering the difference of corresponding curvatures. We will define a subclass of operators of B₁?D? which including the weighted shift operators with weighted sequences {[??+1?/??+2?^?]}??=0, ?? 1 and consider when an operator in 1?D? will be similar to some operator in B₁?D? this operator class.This paper also studies on the generalized Cowen-Douglas operators on Hilbert C*-modules. Hilbert C*-modules to the C*-algebra and the modules structure closely linked,it is the development of the Hilbert space, so study on the properties and the structure of the operators has more research value and significance. This paper proves a typical example of the generalized Cowen-Douglas operators on Hilbert C*-modules, it also studies on the relationship between the backward shift operators in the new inner product and the generalized Cowen-Douglas operators with index one on Hilbert C*-modules.This paper is divided into three parts, each part of the main contents are as follows:In the first part, we introduce some preliminary knowledge which is used in this paper, such as the curvatures of Cowen-Douglas operators, unilateral weighted shift operators and the definition of generalized Cowen-Douglas operators on Hilbert C*-modules.In the second part, we first define a subclass ???_x of Cowen-Douglas operators with index one, then we research when an operator in B₁?D? and the operator ???_x are similar by studying the difference of corresponding curvatures, and obtain an important theorem of this article: Let T ? B₁?D? and let be a holomorphic function on D and also continuous on the closure of the unit disk. For any S? ???_x, if-K_T+K_S = ???|w|²? and?c^??ⁿ?0? > 0,then T^S . According to this ???_x theorem Scontains the weighted shift operators with weighted sequences {[?k+1?/?k+2?^?]}?k=0, ?? 1, based on this theorem also obtained some corollaries and gives proof.In the third part, we prove a typical example of the generalized Cowen-Douglas operators with index one on , get off some conclusions in B₁?D, H_A?, and the different structural effect of modules structure to Cowen-Douglas operators be studied.