Abstract
Assume that \({\mathfrak {X}}\) is a real or complex Hilbert space, T a linear relation in \({\mathfrak {X}}\) and B a bounded linear operator in \({\mathfrak {X}}\), whose adjoints are denoted by \(T^{*}\) and \(B^{*}\), respectively. It is shown in this note that if the following four linear relations \(TBB^{*}T^{*}\), \(B^{*}T^{*}TB\), \(BTT^{*}B^{*}\) and \(T^{*}B^{*}BT\) are selfadjoint in \({\mathfrak {X}}\) then T must be a closed linear relation.
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Roman, M., Sandovici, A. A Generalized Von Neumann’s Theorem for Linear Relations in Hilbert Spaces. Results Math 79, 119 (2024). https://doi.org/10.1007/s00025-024-02145-z
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DOI: https://doi.org/10.1007/s00025-024-02145-z
Keywords
- Hilbert space
- closed linear relation
- nonnegative linear relation
- selfadjoint linear relation
- Von Neumann theorem