Canonical graph contractions of linear relations on Hilbert spaces

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 $P_T^{}{P}_T^{\ast }=(I+T^{\ast }T{)}^{-1}$, and that $Q_T^{}{Q}_T^{\ast }=I-(I+T{T}^{\ast }{)}^{-1}$. The ranges $\mathrm{ran}{P}_T^{\ast }$ and $\mathrm{ran}{Q}_T^{\ast }$ 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.

12 pages