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

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on l 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[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 PhelpsSimons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given. 2000 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47A05, 47B65, 52A41.