Given a densely defined closed operator \(T:{\mathcal {D}}(T)\subset H\rightarrow K\), von Neumann defined \(W:=(I+T^*T)^{-1}\) and showed that \(0\le W\le I\), \({\mathcal {K}}(W)=\{0\}\) and \(T^*T=(I-W)W^{-1}=W^{-1}(I-W)\) with \({\mathcal {D}}(T^*T)={\mathcal {R}}(W)\). (Here, \({\mathcal {D}}(\cdot )\), \({\mathcal {R}}(\cdot )\), \({\mathcal {K}}(\cdot )\) and, later, \({\mathcal {G}}(T)\) stand for the domain, the range, the kernel, and the graph of a linear transformation, respectively.) The functional calculus is not applicable, in general, to guess a formula like \((I-W)^{1/2}W^{-1/2}\) for \(|T|(:=(T^*T)^{1/2})\) and to achieve a polar decomposition \(T=V|T|\) for T. Also, the operators \({\bar{T}}\) and \(T^*\) do not exist as single-valued operators to be able to define W and extend our conjectures to arbitrary unbounded linear operators. The task of the present paper is to define the von Neumann operator W directly from \({\mathcal {G}}(T)\) and prove all the desired extensions.

给定一个密集定义的封闭算子\（T：{\ mathcal {D}}（T）\ subset H \ rightarrow K \），冯·诺依曼定义\（W：=（I + T ^ * T）^ {-1} \）并显示\（0 \ le W \ le I \），\（{\ mathcal {K}}（W）= \ {0 \} \）和\（T ^ * T =（IW）W ^ {-1} = W ^ {-1}（IW）\）与\（{\ mathcal {D}}（T ^ * T）= {\ mathcal {R}}（W）\）。（在这里，\（{\ mathcal {D}}（\ cdot）\），\（{\ mathcal {R}}（\ cdot} \），\（{\ mathcal {K}}（\ cdot} \）然后，\（{\ mathcal {G}}（T）\）分别代表域，范围，核和线性变换的图。）通常，函数演算不适用，猜一个像\（（IW）^ {{1/2} W ^ {-1/2} \）为\（| T |（：=（T ^ * T）^ {1/2}）\）并达到极坐标分解\（T = V | T | \）为Ť。同样，运算符\（{\ bar {T}} \）和\（T ^ * \）不作为单值运算符存在，无法定义W并将我们的猜想扩展到任意无界线性运算符。本文的任务是直接从\（{\ mathcal {G}}（T）\）定义von Neumann算符W并证明所有期望的扩展。