  Monotone operators are of basic importance in optimization as they generalize
simultaneously subdifferential operators of convex functions and positive
semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the
additive decomposition of a maximal monotone operator as the sum of a
subdifferential operator and an "irreducible" monotone operator. In 2007,
Borwein and Wiersma [SIAM J. Optim. 18 (2007), pp. 946-960] introduced another
additive decomposition, where the maximal monotone operator is written as the
sum of a subdifferential operator and a "skew" monotone operator. Both
decompositions are variants of the well-known additive decomposition of a
matrix via its symmetric and skew part.
  This paper presents a detailed study of the Borwein-Wiersma decomposition of
a maximal monotone linear relation. We give sufficient conditions and
characterizations for a maximal monotone linear relation to be Borwein-Wiersma
decomposable, and show that Borwein-Wiersma decomposability implies Asplund
decomposability. We exhibit irreducible linear maximal monotone operators
without full domain, thus answering one of the questions raised by Borwein and
Wiersma. The Borwein-Wiersma decomposition of any maximal monotone linear
relation is made quite explicit in Hilbert space.
