Canonical graph contractions of linear relations on Hilbert spaces (Q2027983)

Let \(\mathcal{H}\) and \(\mathcal{K}\) be Hilbert spaces. Considering a closed linear relation \(T\) as a linear subspace of \(\mathcal{H} \times \mathcal{K}\) the authors call canonical contractions of \(T\) the natural projections \(P_T \colon T \to \mathcal{H}\) and \(Q_T \colon T \to \mathcal{K}.\) The properties of canonical contractions and their adjoints \(P_T^\star,\) \(Q_T^\star\) are studied. In particular, the equalities \(P_TP_T^\star = (I + T^\star T)^{-1}\) and \(Q_TQ_T^\star = I - (I + TT^\star)^{-1}\) are verified. The connection of canonical contractions to the Stone decomposition of a closed linear relation is discussed.