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

给定两个希尔伯特空间\（\ mathcal {H} \）和\（\ mathcal {K} \）之间的闭合线性关系T，相应的第一和第二坐标投影\（P_T \）和\（Q_T \）都为从T到\（\ mathcal {H} \）以及到\（\ mathcal {K} \）的线性收缩。在本文中，我们研究了这些图收缩的特征。除其他外，我们显示\（P_T ^ {} P_T ^ * =（I + T ^ * T）^ {-1} \）和\（Q_T ^ {} Q_T ^ * = I-（I + TT ^ *）^ {-1} \）。范围\（{\ text {ran}} P_T ^ {*} \）和\（{\ text {ran}} Q_T ^ {*} \）被证明与T的所谓“常规部分”密切相关。还讨论了图投影与闭合线性关系的Stone分解的关系。