  In this paper, we study properties of ultramaximally monotone operators. We
characterize the interior and the closure of the range of an ultramaximally
monotone operator. We establish the Brezis--Haraux condition in the setting of
a general Banach space. Moreover, we show that every ultramaximally monotone
operator is of type (NA). We also provide some sufficient conditions for a
Banach space to be reflexive by a linear continuous and ultramaximally monotone
operator.
