  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.
