Invariant Subspaces,Reducing Subspaces And Related Operators On Bergman Spaces

Research of invariant subspaces and reducing subspaces in Bergman space not only closely link to the invariant subspace problem in operator theory, but also closely to many branches of mathematics, such as function theory, Banach algebra. And it has an extensive and profound influence on control,optimization,quantum mechanics and other disciplines.The main work of this thesis is to discuss the invariant subspaces, root operator, re-ducing subspaces and the algebraic properties of Toeplitz operators on Bergman space.We discuss the root operator on weighted Bergman space Aα2(D) of the unit disk, where a is a nonnegative integer. By the reproducing kernel of invariant subspace, root operator is defined. It is calculated that root operator is the sum of some shift operators'products, and several expressions of root operator are obtained. By the relationship between the root operator and the shift operator restricted to invariant subspace, we obtain the spectral properties, and prove that the root operator is compact only when the index of invariant subspace is finite. And we give some examples and estimate ranks of the root operators.We deal with the complete characterization of the minimal reducing subspaces of Toeplitz operators on weighted Bergman space Aα2(D2) over bidisc. Utilizing the orthogonal decomposition of any monomial over bidisc with respect to the reducing subspace, we com-pletely describe the minimal reducing subspaces of Toeplitz operators whose symbols are z1N,z2N,and z1N z2N (N> 1). In weighted Bergman space Aα2(Dn) over polydisc, some similar results of the complete characterization of the minimal reducing subspaces are obtained. Through describing the matrix form of Toeplitz operator TZi(1≤i≤n), Beurling-type theorem is obtained. Consequently, the structure of such invariant subspace is verified.On the pluriharmonic Bergman space of the unit ball, by function theory, we obtain a necessary condition for the product of two Toeplitz operators with pluriharmonic symbols to be a Toeplitz operator, and also characterize (semi-)commuting Toeplitz operators with some pluriharmonic symbols.