Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

TL;DR

This paper provides explicit examples of unbounded linear maximal monotone operators, addressing a recent open question and illustrating limitations in the theory of sums of maximal monotone operators.

Contribution

It introduces two new explicit unbounded linear maximal monotone operators, answering a question by Svaiter and showing constraints on sum maximality conditions.

Findings

The operator S on ℓ² is skew and its domain is a proper subset of its adjoint's domain.

The inverse Volterra operator T on L²[0,1] has a skew part with two distinct maximal monotone skew extensions.

These examples demonstrate limitations in the constraint qualification for sums of maximal monotone operators.

Abstract

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$. We compare the domain of $T$ with the domain of its adjoint $T^*$ and show that the skew part of $T$ admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators can not be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.