Research on the relative position of subspaces of a Hilbert space and operators

研究了Hilbert空间中n个子空间的相对位置。考虑不可分解的位置是很重要的。解决了n=1和n=2的情形。在有限维Hilbert空间中，我们解决了n=3和n=4的情形，并对不可分解的n个子空间进行了完全分类，但在无限维Hilbert空间中，即使是n=3和n=4的情形也没有得到解决。我们开始研究箭图的Hilbert表示，它将Hilbert空间与箭图的顶点和箭头的算子联系在一起。利用动态图A、D和E研究了无限维Hilbert空间中Gabriel定理的一个补集，证明了推广的动态图存在无限维不可分解的Hilbert表示。

We study the relative position of n subspaces of a Hilbert space. It is important to consider indecomposable position. The case of n =1 and n =2 are solved. In finite dimensional Hilbert space, the case of n =3 and n =4 are solved and the indecomposable n subspaces are completely classified.But infinite dimensional Hilbert space, even the case of n =3 and n =4 are still unsolved.In our study, we attacked it by considering an analog of operator theory. We began to study Hilbert representations of quivers, which associate Hilbert spaces and operators for vertices and arrows of quivers. We investigate a complement in an infinite dimensional Hilbert space for Gabriel theorem using Dynkin diagrams A, D and E. We show that there exist infinite dimensional indecomposable Hilbert representations for extended Dynkin diagrams.