A Generalized Von Neumann’s Theorem for Linear Relations in Hilbert Spaces

Assume that \({\mathfrak {X}}\) $\mathfrak{X}$ is a real or complex Hilbert space, T a linear relation in \({\mathfrak {X}}\) $\mathfrak{X}$ and B a bounded linear operator in \({\mathfrak {X}}\) $\mathfrak{X}$ , whose adjoints are denoted by \(T^{*}\) $T^{\ast }$ and \(B^{*}\) $B^{\ast }$ , respectively. It is shown in this note that if the following four linear relations \(TBB^{*}T^{*}\) $TB{B}^{\ast }{T}^{\ast }$ , \(B^{*}T^{*}TB\) $B^{\ast }{T}^{\ast }TB$ , \(BTT^{*}B^{*}\) $BT{T}^{\ast }{B}^{\ast }$ and \(T^{*}B^{*}BT\) $T^{\ast }{B}^{\ast }BT$ are selfadjoint in \({\mathfrak {X}}\) $\mathfrak{X}$ then T must be a closed linear relation.