On the closure of the product and sum of linear relations (Q371691)

Let \(\mathcal H\) be a Hilbert space, \(H=\mathcal H\oplus\mathcal H\), \(\mathcal I\) the identity operator on \(H\) and \(K\) the operator on \(H\) defined by \(K= \left( \begin{smallmatrix} 0 & -i\mathcal I\\i\mathcal I &0 \end{smallmatrix} \right)\). Let \(E,\) \(F\) be two linear relations on \(\mathcal H\) and \(S_{E,F}\), \(T_{E,F}\) be the linear relations defined by \(G(S_{E,F})=G(E)\dot{+}K(G(F)),\) \(G(T_{E,F})=G(E)\dot{+}G(-F),\) where \(G(E)\) and \(G(F)\) denote the graphs of \(E\) and \(F\), respectively. In this paper, the author shows that, if \(E\) and \(F\) are two linear relations, then: {\parindent=6mm \begin{itemize}\item[(1)] if \(S_{E,F}\) is closed, then \(\overline{FE}=(E^*F^*)^*;\) \item[(2)] if \(T_{E,F}\) is closed, then \(\overline{E+F}=(E^*+F^*)^*\). \end{itemize}}