Research of a Hardy space on a polydisc

研究双圆盘上哈代空间H^2（D^2）中不变子空间上的算子.该研究与不变子空间M的结构密切相关。本课题主要研究了以下几个方面。(1)在一定条件下，我们刻画了M的正交补N上的双交换压缩移位可交换的算子。并将其应用于双圆盘上全纯函数的插值问题，证明了Nevanlinna-Pick型定理。(2)我们在M和N中定义游荡子空间。它们分别是M和N的重要小部分。明确了M的游荡子空间与N的游荡子空间之间的关系。(3)考虑两个限制在不变子空间M上的解析Toeplitz算子。当它们的交叉换位子为零时，我们研究M。然后，我们对一些特殊的解析Toeplitz算子完全刻画了M. (4)当Toeplitz算子的不变子空间格包含在解析Toeplitz算子的不变子空间格的交中时，证明了Toeplitz算子是解析的.此外，我们还考虑了C^* 值一元哈代空间及其上的两个移位，给出了C^* 代数在两个不变子空间上酉等价的充要条件.

We study an operator on an invariant, subspace in Hardy space H^2(D^2) on the bidisc. The research is strongly related to the structure of an invariant subspace M. The following are studied in this project.(1) Under some condition, we describe operators which commute with doubly commuting compressed shifts on the orthogonal complement N of M. Moreover applying it to an interpolation problem for holomorphic functions on the bidisc, we prove a theorem of Nevanlinna-Pick type.(2) We define wandering subspaces in M and N. They are important small parts of M and N, respectively. We make clear relations between a wandering subspace of M and one of N.(3) We consider two analytic Toeplitz operators restricted to an invariant subspace M. We study M when the cross commutator of them is zero. Then we describe completely M for some special analytic Toeplitz operators.(4) When the lattice of invariant subspaces of a Toeplitz operator is contained in the intersection of lattices of invariant subspaces for analytic Toeplitz operator, we show the Toeplitz operator is analytic. Moreover we consider C^*-valued one variable Hardy space and two shifts on it. We give necessary and sufficient condition for unitary equivalence of C^* algebras on two some invariant subspaces.