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arXiv · 2007.00954

Canonical graph contractions of linear relations on Hilbert spaces

Zsigmond Tarcsay·Zoltán Sebestyén
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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.

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Zsigmond Tarcsay, Zoltán Sebestyén. 2020-07-02. Canonical graph contractions of linear relations on Hilbert spaces. https://arxiv.org/abs/2007.00954

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