Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Abstract Given a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and {K}$$ K , the corresponding first second coordinate projections $$P_T$$ P T $$Q_T$$ Q are both contractions from to respectively. In this paper we investigate features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ ? = ( I + ) - 1 $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$ . The ranges $${\text {ran}}P_T^{*}$$ ran {ran}}Q_T^{*}$$ proved be closely related so called ‘regular part’ connection Stone’s decomposition is also discussed.