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On the Structure of Unbounded Linear Operators
Iranian Journal of Science and Technology, Transactions A: Science ( IF 1.4 ) Pub Date : 2020-09-03 , DOI: 10.1007/s40995-020-00965-6
Shahrzad Azadi , Mehdi Radjabalipour

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.



中文翻译:

关于无界线性算子的结构

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\(0 \ le W \ le I \)\({\ mathcal {K}}(W)= \ {0 \} \)\(T ^ * T =(IW)W ^ {-1} = W ^ {-1}(IW)\)\({\ mathcal {D}}(T ^ * T)= {\ mathcal {R}}(W)\)。(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\((IW)^ {{1/2} W ^ {-1/2} \)\(| T |(:=(T ^ * T)^ {1/2})\)并达到极坐标分解\(T = V | T | \)Ť。同样,运算符\({\ bar {T}} \)\(T ^ * \)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。

更新日期:2020-09-03
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