An asymmetric Putnam-Fuglede theorem for unbounded operators (Q2718999)

This is a generalization to unbounded operators of the Putnam-Fuglede intertwining theorem where moreover the request of normality is weakened. Let \(A\) be a closed subnormal operator in a Hilbert space \({\mathcal H}\) and \(B\) a closed hyponormal operator in a Hilbert space \({\mathcal K}\). If there exists a bounded linear operator \(X\) from \({\mathcal H}\) to \({\mathcal K}\) such that \(XA^*\subseteq BX\), then \(XA\subseteq B^*X\), \(|X|A\subseteq A|X|\), \(|X^*|B\subseteq B|X^*|\), and the initial and the final space of the partial isometry belonging to the polar decomposition of \(X\) reduce \(A\) and \(B\) to normal operators respectively. The same is true if the properties subnormal and hyponormal are interchanged. The main tool is the following result:NEWLINENEWLINENEWLINEIf \(N\) is a normal operator in \({\mathcal H}\) and \(T\) is a closed hyponormal operator in \({\mathcal K}\) such that \(XN\subseteq TX\) for an \(X\) as before, then \(|X|N\subseteq N|X|\) and \(N= T\) if \({\mathcal K}= {\mathcal H}\), \(X\geq 0\) and \(X\) is injective.