关于Van Daele-Schmüdgen定理
Around the Van Daele--Schmüdgen theorem
摘要：
对于作用在无限维可分复Hilbert空间H中的具有稠密值域R的{有界}非负自伴算子，我们提出了一种新的方法来刻画子空间M \subset H使得M \capR =M^ \perp \capR =\{0\}的存在性现象.我们展示了如何存在这样的子空间导致各种{病态}性质的{无界}自伴运营商有关冯诺依曼定理\cite{Neumann} - \cite{Neumann2}。我们修改了冯·诺伊曼-范·达勒-施米德根的断言\cite{Neumann}、\cite{Daele}、\cite{schmud}，以使它们更加精炼。我们还发展了{一个新的系统方法，它允许对任何{无界}稠密定义的对称/自伴算子T构造无穷多对它的闭稠密定义限制T_k \subset T，使得\dom（T^* T_{k}）=\{0\}（\Rightarrow \dom T_{k} ^2=\{0\}$）k= 1，2且\dom T_1 \cap \dom T_2=\{0\}，\dom T_1 \dot + \dom T_2= \dom T。
英文摘要：
For a {bounded} non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show how the existence of such subspaces leads to various {pathological} properties of {unbounded} self-adjoint operators related to von Neumann theorems \cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van Daele-Schmüdgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to refine them. We also develop {a new systematic approach, which allows to construct for any {unbounded} densely defined symmetric/self-adjoint operator T infinitely many pairs of its closed densely defined restrictions T_k\subset T such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and \dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T.
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