Toeplitz Operator On Hardy Space

In this thesis we mainly discuss the problem of Toeplitz operators on Hardy space. We focus on the characteristics of Fredholm Toeplitz operator on Ordered Group and the reducing subspaces of Tφ. when,In chapter 1, we discuss some related research background,and introduce some basic definitions and symbols. Finally, the significance of the research is presented.In chapter 2, we introduce some basic definitions and properties of the Hardy space and Bergman space.In chapter 3, the characteristics of Toeplitz operator with Fredholm on Ordered group are elaborated.In chapter 4, the reducing subspaces of the Toeplitz operators on Hardy space Tφ, whenφ(z)=B4(z) is studied.