  Let $A$ be a, not necessarily closed, linear relation in a Hilbert space
$\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran
B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus
(\{0\}\times \mul A)$, where the sum is componentwise (i.e. span of the
graphs). This decomposition provides a counterpart and an extension for the
notion of closability of (unbounded) operators to the setting of linear
relations. Existence and uniqueness criteria for the existence of an operator
part are established via the so-called canonical decomposition of $A$. In
addition, conditions are developed for the decomposition to be orthogonal
(components defined in orthogonal subspaces of the underlying space). Such
orthogonal decompositions are shown to be valid for several classes of
relations. The relation $A$ is said to have a Cartesian decomposition if
$A=U+\I V$, where $U$ and $V$ are symmetric relations and the sum is
operatorwise. The connection between a Cartesian decomposition of $A$ and the
real and imaginary parts of $A$ is investigated.
