Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Given a closed linear relation T between two Hilbert spaces H and K , the corresponding first and second coordinate projections P T and Q T are both linear contractions from T to H , and to 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 , and that Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges ran P T ∗ and 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.