Research of a Hardy space on a polydisc

研究了Hardy空间H^2（D^2）上的不变子空间上的一个算子。本课题的研究与不变量子空间m的结构密切相关。(1)在一定条件下，我们描述了在m的正交补N上具有双交换压缩位移的交换算子，并将其应用于二次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上定义游荡子空间，它们分别是M和N的重要小部分。(3)考虑两个解析Toeplitz算子限定在不变子空间M上，研究了它们的交叉换易子为零时M的性质。然后对一些特殊的解析Toeplitz算子进行了完全的描述。(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集中时，证明Toeplitz算子是解析的。此外，我们还考虑了C^*值的一元Hardy空间及其上的两次移位。给出了两个不变子空间上C^*代数酉等价的充分必要条件。

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.