Canonical graph contractions of linear relations on Hilbert spaces

TL;DR

This paper studies the properties of graph contractions associated with closed linear relations on Hilbert spaces, revealing their algebraic structure and connections to the regular part and Stone's decomposition.

Contribution

It provides new insights into the structure of graph contractions of linear relations, including explicit formulas and their relation to the regular part and Stone's decomposition.

Findings

Derived formulas for $P_T^{}P_T^*$ and $Q_T^{}Q_T^*$.

Linked the ranges of $P_T^*$ and $Q_T^*$ to the regular part of $T$.

Connected graph projections to Stone's decomposition.

Abstract

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.