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 $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$
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.
Linear Relations in Hilbert Spaces
