Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Zsigmond Tarcsay, Zoltán Sebestyén
2021 Complex Analysis and Operator Theory  
AbstractGiven a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and $$\mathcal {K}$$ K , the corresponding first and second coordinate projections $$P_T$$ P T and $$Q_T$$ Q T are both linear contractions from T to $$\mathcal {H}$$ H , and to $$\mathcal {K}$$ K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ P T P T ∗ = ( I + T ∗ T ) - 1 , and that
more » ... {}Q_T^*=I-(I+TT^*)^{-1}$$ Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges $${\text {ran}}P_T^{*}$$ ran P T ∗ and $${\text {ran}}Q_T^{*}$$ 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.
doi:10.1007/s11785-020-01066-3 fatcat:xuz7nhzci5bw3lj4rfz6jsh7my