# On Borwein-Wiersma Decompositions of Monotone Linear Relations

## TL;DR

This paper investigates the Borwein-Wiersma decomposition of maximal monotone linear relations, providing conditions, characterizations, and explicit forms, and explores the relationship with Asplund decomposability.

## Contribution

It offers new insights into Borwein-Wiersma decompositions, including conditions for decomposability and examples of irreducible operators, advancing understanding of monotone linear relations.

## Key findings

- Borwein-Wiersma decomposability implies Asplund decomposability.
- Explicit form of the decomposition in Hilbert spaces.
- Existence of irreducible operators without full domain.

## Abstract

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.