Algebraic Properties Of Small Hankel Operators And Toeplitz Operators With Quasihomogeneous Symbols On The Harmonic Bergman Space

In this dissertation, it studies the algebraic properties of small Hankel operators and Toeplitz operators with quasihomogeneous symbols on the harmonic Bergman space.Firstly, the commuting of small Hankel operators with quasihomogeneous sym-bols is considered and then, a complete characterization for commutativity of two small Hankel operators with symbols one is quasihomogeneous and another bounded is given, the result obtained here is quite different with the similar problem on the other function space cases. Moreover, the commuting of small Hankel operator and Toeplitz opera-tor with quasihomogeneous symbols is also considered. Secondly, when the product of small Hankel operator and Toeplitz operator with the radial symbols being another small Hankel operator is answered. At last, it studies the finite rank problem of finite product of small Hankel operators or commutator of small Hankel operator and Toeplitz operator with quasihomogeneous symbols, and the obtained results show that they hold are only the trivial ways.