Properties Of Toeplitz Operators On The Harmonic Bergman Space

As an important branch in operator theory, operator theory in function spaces not only closely links to many areas of mathematics, but also closely to other disciplines.In the last decade, properties of Toeplitz operators have been fully studied on analytic Bergman space La2, including boundedness, compactness and algebraic properties. However, The theory of Toeplitz operators on the harmonic Bergman space Lh2is quite different from that on La2. Properties of Toeplitz operators on the harmonic Bergman space are difficult to characterize, because of conjugated basis. The content of this paper are as follows:In chapter1, we introduce the development and current situation of the research on oper-ator theory in function spaces, including Toeplitz Operator theory in classic Hardy space H2, Bergman space and the harmonic space.In chapter2, we mainly introduce definitions and properties of Bergman space L2and Toeplitz operators on it. We firstly gave necessary and sufficient conditions for the product of two Toeplitz operators with harmonic symbols to be a Toeplitz operator on La2, and explained the same question of Toeplitz operator with quasihomogenuous symbols. Then verified the products of operators Tz2, Tz2, TzTz, TzTz and7z+z2are Toeplitz operators.In chapter3, we firstly introduce definitions and properties of the harmonic Bergman space Lh2and Toeplitz operators on it. then characterize all Toeplitz operators which commute with Tz If Toeplitz operators with quasihomogenuous symbols could commute with Tz, their symbols and one of z, z-1or the constant function1are linearly dependent. Later, through operating on the basis of the harmonic Bergman space, we show the products of operators Tz2, Tz2, TzTz, TzTz and TzTz+TzTz are not Toeplitz operators. Lastly, we discuss the polar decomposition of Tz+z2, and prove it isn’t a Toeplitz operator.