On the Structure of Unbounded Linear Operators

Abstract

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.