# Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter - Mathematics > Functional Analysis

Abstract: In this paper, we give two explicit examples of unbounded linear maximalmonotone 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 itsadjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answerto a recent question posed by Svaiter. The second unbounded linear maximalmonotone operator is the inverse Volterra operator $T$ on $L^{2}0,1$. Wecompare the domain of $T$ with the domain of its adjoint $T^*$ and show thatthe skew part of $T$ admits two distinct linear maximal monotone skewextensions. These unbounded linear maximal monotone operators show that theconstraint qualification for the maximality of the sum of maximal monotoneoperators can not be significantly weakened, and they are simpler than theexample given by Phelps-Simons. Interesting consequences on Fitzpatrickfunctions for sums of two maximal monotone operators are also given.

Author: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao

Source: https://arxiv.org/