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.

在本文中，我们首先研究具有有限缺陷的标量和向量值近不变子空间的结构。然后，我们随后对我们的新结果进行一些卓有成效的应用。我们针对具有有限缺陷的向量值近不变子空间产生分解定理。更具体地说，我们显示了具有有限缺陷的每个向量值几乎不变的子空间可以写为向后移动不变子空间的等轴测图。我们还表明，向量值几乎不变的子空间与标量值几乎不变的子空间之间存在联系，并且存在有限缺陷。这是一个有力的结果，使我们能够使用向量值的Hardy空间技术深入了解Hardy空间的标量子空间的结构。这些结果具有深远的应用价值，