Cowen-Douglas Operators And Shift On Basis

In the recent years, the development of bases theory and operator theory on Hilbert space and Banach space is very fast. Also, a growing number of researchers are added to the study of these problems. All the problems we talk about are on Hilbert space. M. J. Cowen and R. G. Douglas have done very well in the connection between the operator T in Hilbert space and complex bundle ET. But now we still have not exactly established contact Hilbert space with shift operator T. In this paper, we will discuss the problem plainly.In the first part,We will give some information about the basic definitions used in the paper at the beginning of the introduction,It will provide the basic basis for the following argument, Then three theorems of Cowen-Douglas operator in two orthogonal systems are presented.In the second part,we shall recall some basic def-initions and theorems about the cross-section on Cowen-Douglas operator complex bundle.The main theorems 1.1 will be proved in the third part.Then we will talk about the shift of Cowen-Douglas operator on biorthogonal system in lemma 3.2 and lcm-ma 3.3, and then we prove Theorem 1.2 and Theorem 1.3. In the last part, we focus on B1(Ω). Theorem 4.1 gives the equivalent condition of whether an operator T in B1(Ω) is a shift on some Markushevicz basis. Lastly, Theorem 4.2 gives the operator theory description about the condition of an operator T in B1(Ω) being the backward weighted shift on orthogonal basis.