Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators
Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators
复制标题
近不变子空间及其在截断托普利茨算子中的应用
DOI：
10.1007/s11785-020-01049-4
复制
发表时间：
2020
影响因子：
0.8
通讯作者：
O'Loughlin R
中科院分区：
文献类型：
--
作者：
O'Loughlin R
本文首先研究了具有有限亏量的标量和向量值几乎不变子空间的结构。然后，我们随后对我们的新结果进行了一些富有成效的应用。给出了具有有限亏损的向量值几乎不变子空间的一个分解定理。更具体地说，我们证明了每个具有有限亏损的向量值几乎不变子空间都可以写成一个向后移位不变子空间的等距像。我们还证明了向量值几乎不变子空间与具有有限亏量的标量值几乎不变子空间之间存在联系。这是一个强有力的结果，它允许我们使用矢量值Hardy空间技巧来深入了解Hardy空间的标量子空间的结构。这些结果有着广泛的应用，特别是它们允许我们发展一种全方位的方法来研究Toeplitz算子、截断Toeplitz算子、多带空间上的截断Toeplitz算子和对偶截断Toeplitz算子。
In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.