The Toeplitz Operator On Harmonic Bergman Space And Hardy Space

The thesis consists of four parts. It mainly investigates some problems on harmonic Bergman space and Hardy space.Chapter 1. Introduction and preliminary knowledge. We concisely introduce some elementary theorem which we will use in the thesis.Chapter 2. The necessary and sufficient condition for the weak convergence on harmonic Bergman space. We present the necessary and sufficient condition for the weak convergence of function sequence on harmonic Bergman space, the conclusion is similar to the case of Bergman space, but some differences on the prove process. Meanwhile, we discuss the compactness of Toeplitz operator with some symbols on Bergman space, these results extend the proposition 6.1.6 in [10], which is different on the method, at the same time, we transit it to harmonic Bergman space.Chapter 3. The bounded projection of L~p(D,dA) onto L_h~p(D). The result extends the case of Bergman space in [9].Chapter 4. The short exact sequence on Hardy space and the application. We give three short exact sequence on Hardy space, it mainly involve the Toeplitz operator on Hardy space, and we overcome the difficulty of the lack of involution operation on H~p.