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Canonical graph contractions of linear relations on Hilbert spaces

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Zsigmond Tarcsay, Zoltán Sebestyén

Functional Analysis · Submitted 2 Jul 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show among others that $P_T^{}P_T^*=(I+T^*T)^{-1}$, and that $Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$.

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arXiv ID2007.00954 (version 1)
Pith IDDXF53FQ3, Pith's permanent record of this paper and its review
CitationZolt\'an Sebesty\'en, Zsigmond Tarcsay. "Canonical graph contractions of linear relations on Hilbert spaces." arXiv:2007.00954 (2020). https://pith.science/paper/DXF53FQ3
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arXiv listing abstract v1. Results and the main claim are underlined; .

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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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. background 0.96 0.20 0.06 asserted 1.00
s1 In this paper we investigate the features of these graph contractions. contribution 0.98 0.59 0.04 asserted 0.63
s2 We show among others that $P_T^{}P_T^*=(I+T^*T)^{-1}$, and that $Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$. result 1.00 0.94 0.34 asserted 1.00
s3 The ranges $\operatorname{ran} P_T^{*}$ and $\operatorname{ran} Q_T^{*}$ are proved to be closely related to the so called `regular part' of $T$. result 0.98 0.90 0.17 asserted 1.00
s4 The connection of the graph projections to Stone's decomposition of a closed linear relation is also discussed. contribution 0.93 0.65 0.07 asserted 0.81

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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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