Unitary Constant Regeneration Of Analytic Hilbert Modules Over Operator Theory

This thesis deals mainly with some basic structures of the analytic Hilbert modules endowed with unitary invariant reproducing kernel, denoted by Hgd for short, and operator theory on these Hilbert modules. In particular, some basic problems of operator theory, such as the standard dilation theory of the d-contraction on Hgd and the associated von Neumann inequality, the Toeplitz C*-algebra and its automorphism group, the boundedness and the compactness of the composition operators over Hgd are discussed. In addition for d=l, a kind of semi-order structure for Hgl(C) is characterized, and some basic properties of composition operators on Riemann surface are also given in this thesis.Chapter 1 is concerned in the basic structure and properties of the analytic Hilbert module Hgd(B) with unitary invariant reproducing kernel. For such a Hgd(B), the generating function g is characterized, an orthonormial basis is explicitly given too, by these, some important properties of Hgd(Bd), as well as the relations between the space Hgd(Bd ) and the weighted Arveson space are also discussed.Chapter 2 is devoted to investigating the dilation theory of d-contraction via Hgd(Bd) and the von Neumann inequality. The conditions for multiplication operators to be bounded and essential normal, as well as for the von Neumann inequality to be true, are given respectively.Chapter 3 is concentrated at studying the Toeplitz C*-algebra and its automorphism group. It is shown in this chapter that there is a short exact sequence as followingwhere Tgd and K are the associated Toeplitz C*-algebra and the compact operators respectively. Therefore the Calkin algebra Tgd/K. is isomorphic to C(Bd[s,t]). It is significient here that the maximal ideal space of Td jK. is homeomorphic to a ball-shell which is essential different from all known classical case. Also the index formula for Fredholm Toeplitz d-tuple, as well as the automorphism group of the Toeplitz C*-algebra Tgd are completely determined.Chapter 4 is aimed to investigate the composition operator on Hilbert module Hgd(Cd) and a semi-order relation denned on Hgl(C). It is proved in this chapter that if the composition operator C on Hgd{Cd) is bounded then (z) = Az + 6, with ||A|| < 1: if the composition operator C on Hgd(Cd) is compact then (z) = Az + 6, with ||A|| < 1. Also some criteria for Hgl(C) being or not being semi'-ordered are given in terms of the number is the coefficient sequence of the power expansion of the generating function g(-). In fact it is shown that if l = + then Hgl (C) is of semi-ordered; if l = 0 then Hgl (C) is not of semi-ordered. In the case 0 < l < +, two examples are given to demonstrate that Hgl (C) could be of semi-ordered or opposite.Chapter 5 is about the composition operator on Riemann surface. The criteria for composition operator Cp being normal or quasi-normal are given by counting function Np respectively. It is also shown that if the Riemann surface M is of finite triangulation then many properties of the composition operator Cp, such as normality, quasi-normality, isometricality, invertibility and unitary, are all equivalent to that the map p : M M is a holomorphic, homeomorphism.