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

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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.