论文标题
希尔伯特空间线性关系的规范图收缩
Canonical graph contractions of linear relations on Hilbert spaces
论文作者
论文摘要
鉴于两个Hilbert Spaces $ \ Mathcal H $和$ \ MATHCAL K $之间的封闭线性关系$ t $，相应的第一和第二坐标投影$ p_t $和$ q_t $都是线性收缩。在本文中，我们研究了这些图形收缩的特征。我们表明，$ p_t^{} p_t^*=（i+t^*t）^{ - 1} $，以及$ q_t^{} q_t^*= i-（i+tt^*）^{ - 1} $。范围$ \ operatorName {ran} p_t^{*} $和$ \ operatatorName {ran} q_t^{*} $被证明与$ t $的$ t $的“常规部分”密切相关。还讨论了图形投影与Stone分解封闭线性关系的连接。
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 others that $P_T^{}P_T^*=(I+T^*T)^{-1}$, and that $Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$. The ranges $\operatorname{ran} P_T^{*}$ and $\operatorname{ran} Q_T^{*}$ 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.
